v> 


^\ 


Digitized  by  the  Internet  Archive 

in  2008  with  funding  from 

IVIicrosoft  Corporation 


http://www.archive.org/details/briefcourseinadvOOvelzrich 


A     BRIEK     COURSE     IN 

ADVANCED  ALGEBRA 

Being  Course  Two  in  Mathematics  in  the 
University  of  Wisconsin: 

CONTAINING 


I.     Imaginaries. 

II.     Discussion   of   the    Rational   Integral 
Function   of    x. 

III.  Solution   of   Numerical    Equations    of 

Higher   Degree. 

IV.  Graphic  Representation   of  Equations. 
V.     Determinants. 


C.  A.  VAN   VELZER  and  CHAS.  S.  SLIGHTER. 


EV.-VNSTON,  ILL.: 

UNIVERSITY     PRESS. 


-.i/X^ 


ADVANCED     ALGEBRA. 


CHAPTER     I. 


IMAGINARIES. 


I.  Let  us  notice  what  is  meant  by  Algebraic  Quantity,  and 
how  the  notion  of  it  may  originate.  The  primitive  conception  of 
number  is  used  when  we  enumerate  the  marbles  in  a  box,  and  say: 
o,  I,  2,  3,  4,  etc.  Our  simple  scale  \s,  arithmetical  quantity ,  and 
it  runs  down  to  a  definite  nothing,  or  zero,  and  stops.  But  let  us 
attempt  to  apply  this  scale  in  the  measurement  of  other  things. 
Suppose  we  are  estimating  time.  Where  is  the  zero  from  which 
all  time  is  to  be  measured  in  one  direction,  or  sense?  There  is  no 
such  zero,  as  in  the  case  of  the  marbles;  for  you  can  conceive  of 
no  event  so  far  past  but  what  other  events  preceded  it.  We  are 
forced  to  select  a  standard  event,  and  measure  the  time  of  other 
events  with  reference  to  the  lapse  bifore  or  afte7'  that.  The  zero 
used  is  an  arbitrary'  one,  and  there  is  quantity  in  reference  to  it  in 
two  opposite  senses,  future  and  past,  or,  as  is  said  in  algebra, 
positive  and  negative.  We  are  likewise  obliged    to  recognize 

quantity  as  extending  both  ways  from  a  zero  in  the  attempt  to 
measure  many  other  things;  in  locating  points  along  an  east  and 
west  line,  no  point  is  so  far  west  but  what  other  points  are  west 
of  it,  hence  could  not  be  looted  in  the  arithmetical  scale;  the 
same  in  measuring  force,  which  may  be  attractive  or  repulsive; 
or  motion,  which  maj-  be  toward  ox  from;  etc.  Thus  our  notion  of 
algebraic  quantity  as  we  name  this  kind  of  number,  has  arisen. 

Because  of  the  peculiar  analogy  between  our  notion  of  time 
•and  of  algebraic  qunntity,  algebra   has  been  called  the  science  of 


2  ADVANCED     ALGEBRA. 

pure  time.  All  quantities  are  measured  exactly  as  a  past  and 
future,  or,  graphically,  along  a  line,  in  both  directions  from  a 
zero  point.  But,  with  algebraic  numbers,  we  never  get  out  of  the 
line.  This  kind  of  quantity,  although  more  general  than  arith- 
metical number,  is  really  quite  restricted.  We  observe,  at  once, 
that  there  is  an  opportunity  of  enlarging  our  conception  of  quan- 
tity if  we  can  only  get  out  of  our  line,  or  one  way  spj'ead  as  .some 
say,  and  explore  the  region  without.  We  may  expect,  then,  an 
extension  of  our  notion  of  quantity  which  will  enable  us  to  con- 
sider, along  with  the  points  of  our  Hue,  those  which  lie  without. 

2.  It  is  interesting  to  note  that  any  number  may  be  regarded 
as  the  symbol  of  an  operation;  and  that  thereby  some  original  con- 
ceptions may  be  conveniently  extended  so  as  to  contain  a  meaning 
of  greater  range.  Thus  lo  may  be  regarded,  not  only  as  feji,  but 
as  denoting  the  operation  of  taking  unity  ten  times  ;  to  express 
this,  we  may  write  lo  i.  In  the  same  way,  5  denotes  the  opera- 
tion of  taking  unity  five  times.  Now  we  notice  that  an  exponent 
will  have  an  extended  significance.  It  means  to  repeat  the  opera- 
tion designated;  i-  e.,  the  operation  designated  is  to  be  performed, 
and  performed  again  on  the  result  and  performed  again  on  this 
result ;  and  so  on,  equal  to  the  number  of  times  told  by  the  expo- 
nent. Thus  10''  means  to  perform  the  operation  of  repeating 
unity  ten  times  (told  in  10)  and  then  to  perform  the  operation  of 
repeating  this  result  ten  times,  ?.  e.,  loUo-i).  Also  lo'^  means 
io[io(ioi)].  Then,  of  course,  the  exponent  zero  can  only  mean 
that  the  operation  on  unity  denoted  by  the  quantity  is  not  to  be 
performed  at  all;  i.  e.,  unity  is  to  be  left  unchanged;  thus  10"  or 
io"-i  =  i.  An  expression  hke  v/  4  ,  looked  upon  as  a  symbol  of  an 
operation  denotes  an  operation  such  that  if  the  operation  were  to  be 
performed  twice  the  result  would  be  the  same  as  is  indicated  by  the 
expression  under  the  radical  itself;  /.  e.,^'~^  stands  for  the  opera- 
tion which  must  be  repeated  to  be  equivalent  to  quadrupling, 
which  is  indicated  by  the  expre.ssion  uader  the  radical  (4);  hence 
y/  4    represents  the  operation  oi  doubling.     The  expre.ssion — 1  as 


IMAGINARIES.  3 

a  symbol  of  operation  is  a  reverser,   denoting   the   operation  of 
changing  from  one  sense  to  the  opposite  sense. 

3.  We  remember  that  such  expressions  as  3 +  \/^I^,  <:+\/^, 
\/^,  etc.,  were  forced  upon  our  notice  in  the  solution  of  quad- 
ratic equations.  It  is  customary  to  call  such  Imaginaries,  be- 
cause of  the  presence  in  them  of  a  term  like\/^  which,  evidently, 
does  not  correspond  to  any  algebraic  quantity  whatever.  But,  re- 
membering the  restricted  nature  of  algebraic  quantity,  it  is  possible 
that  such  expressions  are  unreal  only  in  an  algebraic  sense;  that 
if  the  restriction  can  be  removed  by  an  extension  of  our  concep- 
tion beyond  a  mere  linear  or  past-and-future  notion  of  quantity, 
the  expression  may,  perhaps,  become  as  real  to  us  as  algebraic 
numbers  now  are. 

While  v/^  is  not  an  algebraic  number,  yet  we  may  give  to 
it  any  definition  as  a  symbol  of  operation  which  is  consistent  with 
this  one  condition:  (v/^)'= — i;  that  is,  two  of  the  opera^ons 
must  be  a  reversal..  So,  as  — i  may  be  defined  as  such  a  symbol 
which  operates    to  turn    a   straight  b 

line  through  iSo°,  in  a  similar  way  ^ 

we  make  this  1  7 

4.  Definition.     Thisexpres-  "^ 


sion  \/^  is  defined  as  that  symbol  (-    (^zz'ya   O 
which  denotes  the  operation  of  turn-  ^^^ 

ing  a  straight  line  through  an  angle 


A 


iT 


of  90°  in  the  positive  direction.      It  |  7 

is  customary  in  mathematics  to  con-  "^ 

si  Jer  rotation  opposite  to  that  of  the 
hands  of  a  watch  as  positive  rotation. 

In  the  figure,  let  a  beany  line.  Then  a  operated  on  by^^, 
/.  <?.,  -v/^-a  is  a  turned  ?<!/>,  or  positively,  through  90°.  Now,  of 
course,  >/— i  can  operate  on  y/~i  a  just  as  well  as  on  a.  Then 
■v/— i(-v/— i^)  is  v/— I  turned  positively  through  90°; 
v^— |[n/— i(v/— I  «";]    is    ^~i{y/^.a)   turned    through    90°;  etc. 

As  we  are  at  liberty  to  consider  two  turns  of  90°  as  the  same 
as    one  turn    of     180°,  .•.    v/_,(\/^^)  =  (— 1)«.       Also    nn= 


4  ADVANCED     ALGEBRA. 

(—i)OB,.-.  On=— iV—ra),  hut  s/^i{- a  )=  on,  .-.— (^"i-a) 

^■<y—i( — a).     Thus  the  student  may  show  many   like  relations. 

5.  In  order  that  imaginaries  may  be  conveniently  used  in 
algebraic  work  the  following  useful  conventions  governing  the  or- 
dinary algebraic  operations  with  them  are  adopted: 

a.  The  Associative  Lazv,  by  which  the  result  of  a  repeated 
operation  is  to  be  regarded  the  same  irrespective  of  the  mode  of 
grouping  the  elements,  so  long  as  their  order  is  not  disarranged. 
That  is,  {c+s/'^vd)-\-{e-\-^^/)  means  the  same  as 

c+iV^^id+o  +  V^if.     Also.  {^—x-a){b)  =  {s/'^i){ab). 

b.  That  s/^^i  a  is  to  mean  the  same  as  v/^^  and  vice  versa. 

c.  The  Commutative  Law,  by  which  the  result  of  several 
operations  is  to  be  regarded  the  same  irrespective  of  the  order  in 
which  they  are  performed.  That  is,  ^v/ZTf  means  the  same  as 
y/^{  a  ;  s/^^v/I^  means  v/^\/— i  «  ;  c-\-ds/^^\-\-e-\-f\/^^\^ 
c-\-e-\-ds/^\-\-fs/^^\.  While  it  seems  more  natural  to  write 
■v/^  «  than  as/—\,  because  of  the  definition  of  \/—\,  yet  it  is 
the  universal  practice  to  write  the  latter.  Of  course  the  equation 
iOv/^  =  -s/^-io,  or  better  iov/^i  =  n/^  10  i,  means  that  the 
resjilt  0/  performing  the  operation  of  turning  unity  through  go° 
and  performing  upon  the  7'esult  the  operation  of  taking  it  ten 
times  is  the  same  as  the  result  of  performing  the  operation  of  taking 
luiity  ten  times  and  pcrformiyjg  upon  this  result  the  operation  of 
turning  thro2cgh  go° . 

d.  The  Distributive  I^aw  in  the  case  of  an  operation  per- 
formed upon  a7i  aggregate,  by  which  the  resul*;  of  such   operation 

upon  the  aggregate  of  several  terms  is  to  be  regarded  the  same  as 
the  aggregate  of  the  results  of  the  operation  upon  each  term  sep- 
arately.    That  is,  (a-\-b)s/^  means  the  same  as  a\/^+b\/^. 

We  agree  by  the  above  that  any  operation  upon  imaginaries 
or  by  imaginaries  is  to  be  regarded  as  associative,  commutative 
or  distributive,  if  it  would  be  so  regarded  when  the  imaginaries 
are  replaced  by  ordinary  quantities,  and  not  otherzuise. 

6.  W^e  will  now  interpret  the  powers  of  s/^  by  means  of 
the  new  significance  of  an  exponent  and  by  the  above  convention -^. 


w 

lAGINARIES. 

(^/= 

ir 

or  is/- 

=7)"  I 

=  -ri 

(s/- 

i)^ 

or  (^/I: 

T)'-i 

=  s/~i 

=  2 

(V= 

T)'-' 

=— I 

=={' 

(^/= 

7)-' 

=   Cv/- 

i)-^  v/= 

T     — ^/- 

-,=t' 

(s/- 

7)^ 

=   (n/= 

T)'-'(n/= 

T)-^  =  +  i 

=  i' 

(x/- 

T)-"' 

=   (n/= 

0*  V= 

7    =n/=T 

=  l' 

(v/= 

I)" 

=   (v/= 

~lf(K/- 

I)'=— I 

=  f' 

(v/= 

0' 

=   (n/= 

T/n/- 

~ /~ 

'  —i' 

(v/= 

'iT 

=  {%/= 

DXn/- 

i/  =  +  i 

=  f 

etc. 

etc. 

etc. 

etc. 

Art. 


15- 


Whence  it  is  seen  that  all  even  powers  of  \/^r  are  either+ 1 
or  — I  and  all  odd  powers  either  \/^i  or  — V^.  Reconcile  this 
with  the  figure  in  Art.  4. 

7.     Theorem.  Any  polynomial  of  the  form 

can  be  put  in  the  typical  fonii,  a  +  d^/^. 

The  symbols  c,,,  r,,  c,,  etc.,  stand  for  any  coefficients,  and  n  is 
a  whole  number. 

(i.)  As  a  particular  case  take 

c,(s/~iy+c^(s/-iy+cXs/~if-^r.js/-iy+cy-i+c,= 

which,  (by  Art.  5,  c)  equals  c^ — r.j+c+r,,^/-^ — c.^ZTx+c^sZ—i, 
which  (by  Art.  5,  d)  equals  {c^ — '^:i+fO  +  (''o — ^2+0^/— i-  which  is 
of  the  form  a-\-b^--\. 

(2.)  In  any  case  the  odd  powers  of  v/^  will  be  either  v/^ 
or — V^;  the  even  powers  will  be  +1  or  — i,  which  are  real. 
Hence  the  result  may  be  expressed  as  an  aggregate  of  real  quan- 
tities and  an  aggregate  of  quantities  operated  on  by-v/^,  which 
is  an  expression  of  the  form  a-\-bs/^,  by  Art,  5,  c  and  d. 

Q.  E.   D. 


ADVANCED     ALGEBRA. 


8.  Any  imaginary,  in  the  typical  form,  may  be  taken  as  the 
representation  of  the  position  of  a  point  in  a  plane.  For,  suppose 
c+^v/HT  is  the  imaginary.  Let  O  be  the  zero  point  and  OX  the 
positive  direction.  Lay  off  OA=+c,  and  at  A  erect  d>/^.  Then 
c-\-dy/—\  defines  the  position  of  the  point  Pwith  reference  to  O. 
Then  OA-\-AP,  or  OP,  is  a  geometrical  representation  of 
c-\-d'y—\.  In  the  same  manner  let  c—d^^,  — c—dy/~\,  and 
—c-\-d>/—\  be  con.structed. 

9.  The  meaning  of  some  of  our  conventions  may  now  be  il- 
lustrated.    Let  us  construct  c-\-dy/—i-\-c-{-fs/—i- 

e  c 


1 

0 

F 
I                            H 

G 
B 

T 

e 

P 

c            / 

1  T 

El 

i 

rf               JD                                    X 

^ 

c 

i 

IMAGIXARIES.  7 

The  first  two  terms,  c-\-dy/—i,  give  OA  +  AB,  locating  B. 
The  next  two  terms,  c+J\/—i,  give  BC+CP,  locating  P.  Hence 
the  entire  expression  locates  the  point  P  with  reference  to  O. 
Now  if  the  original  expression  be  changed  in  any  manner  allowed 
by  our  conventions,  the  result  is  only  a  different  path  to  the  same 
point.     Thus: 

c+c+dV-i+fV-i   is  the  path   OA,  AD,  DC,  CP. 
{c+e)  +  [d-\-f)K/—i    is  the  path   OA,  AD,  DP. 
c^d^-x+c+tV-i   is  the  path   OE,  EH,  HC,  CP. 
e+d^'^i+/s/~+c  is  the  path   OE,  EH,  HE,  EP,  etc. 
The  student  should  consider  other  cases. 

Query.  Arc  there  any  methods  of  locating  P  with  the  same 
four  elements,  which  the  figure  does  not  illustrate? 

10.  Definition.  Two  imaginaries  are  said  to  be  Conju- 
gate when  they  differ  only  in  the  sign  of  the  term  containing 
^_j.     Such  are  c-\-ds/—\  and  c  —  d^—i. 

11.  Theorem.  Conjugate  imaginaries  have  a  real  sum 
and  a  real  product. 

For,  r+  ds/—\  +  c  —dy/^i 

=  c+c+d^:^i—d^-i.  Art.  5,  c. 

=  2c+{d-d)^~i  =  2c  _  Art.  5,  d. 

And  {c-\-ds/^i){c—  dy/^),  by  distributing  the  second  paren- 
thesis through  the  first,  (Art.  5,  d)  equals 

c{c -  dV~i )  +  dV^i  ( c  -  d>/—i ) 
=c--cd^~i_  +  dy/~i  c-ds/^^d^^i   Art.  5,  d. 
=r-\-d'+eds/^~cds/^.  Art.  5,  c. 

=r  +  d'+{ed—cd)s/—i=r^d'  Art.  5,^/. 

Q.  E.  D. 

12.  Theorem.  7 he  sum,  product,  or  quotient  of  tii'o  imagi- 
naries is,  in  general,  an  imaginary  of  the  typical  form , 

I,et  the  two  imaginaries  be  c-\-d^y~\  and  e-\-f\/—\. 
a.     Their  sum  is    {c-\-ds/~\)-\-{e^f^/~\) 

=  { r+  e)  +  ( ds/—i  +/\/~i )  Art.  5 ,  r  and  a . 
=  (^-f^)-f  («'+/)  v/-i  Art.  5,  d. 

which  may  be  written  a-\-b\/^\. 


S  ADVANCED     ALGEBRA. 

b.  The  product  is  {c+ds/—Y){e+fV—i) 

=  ce + c/^—i + ed^—i + df^—i  ^~i  Art.  5 ,  d  and  c. 

=  (ce—d/)  +  (c/-\-ed)^—i  Art.  5,  a  and  d. 

which  may  be  written      a-^-d-,/—!. 

c.  Their  quotient  is 

c->rd^-i_  (r+^v/-i)(g-A/-i) 
^+/"x/-i     {e+jV-x){e-fV-i) 
By  the  preceding,    the  numerator  is  of  the  form   a'-\-b'\/—\. 
By  Art.    11,   the  denominator  equals  e' -{-/'.     Then  the  quotient 


equals 


^'"^^'^-^  ^'      +^A  Art.  5,^. 


which  is  of  the  form  a-\-b\/—i.  Q.  E.  D. 

13.  ThEuREI.i.  If  an  imaginary  is  equal  to  zero,  the  imagi- 
nary and  real  part  are  separately  equal  to  zero. 

Suppose  <a;  +  i^v/— 1=0 
then  a=  —  bs/—\. 
Now  it  is  absurd  tor  a  real  quantity  to  equal  an  imaginary, 
except  they  each  be  zero. 

Therefore  ^-=0  and  <^=o.  Q.  E.  D. 

14.  Corollary.  If  two  iviaginaries  be  equal,  then  the  real 
and  the  imaginary  pa7^ts  must  be  respectively  equal. 

If  f+rt'v/— i=^+A/^i,  then  {c—e)^-{d-f)V-\=o  and 
c—e=o  and  (f— y"=o. 

15.  For  the  purpose  of  abbreviation,  it  is  customarj^  to  rep- 
resent the  symbol  V—i  by  i.  Then  the  typical  imaginary  is 
a  -f  bi.  But  the  student  must  keep  clearly  in  mind  the  meaning 
of  i,  and  also  the  meanings  of  the  powers  of  z.    See  Art.  6. 


CHAPTER   II. 

The  Rational  Integral  Function  of  x. 

1.  Definition.  A.  Variable  is  a  quantity  which  changes 
vaUie.  Variables  are  represented  by  the  last  letters  of  the 
alphabet. 

2.  Definition.  A  Constant  is  a  quantity  which  has  a  fixed 
value.  Constants  are  represented  by  the  first  letters  of  the  alpha- 
bet and  by  numbers. 

3.  Definition.     A  Function  of  a  quantity  is  a  name  applied 

to  any   mathematical   expression   in   which  the  quantity  appears. 

I — ,r  

2ax;       x-y'x';  .        ;        s/a-.v'. 

These  are  all  functions  of  x.  In  the  same  manner  we  speak  of 
functions  of  several  quantities.  The  second  expression  above 
may  be  called  a  function  of  .r  andj'.  Obviously,  a  function  of  a 
quantity  might  be  otherwise  defined  as  any  expression  which  de- 
pends upon  the  quantity  for  its  value. 

4.  Definition.  A  R.ational  Function  of  a  quantity  is  one 
in  which  the  quantity  is  not  involved  in  a  radical,  or  affected  with 
an  irreducible  fractional  exponent. 

5.  Definition.  An  Integral  Function  of  a  quantitj^  is  one 
in  which  the  quantity  does  not  appear  in  the  denominator  of  a 
fraction,  or  is  not  affected  with  a  negative  exponent. 

6.  A  function  may  be  both  rational  and  integral 
in  which  case  it  is  called  a  Rational  Integral  Function 
This  is  the  function  we  propose  to  consider,  but,  while  limit 
ing  ourselves  to  the  consideration  of  this  class  of  func 
tions,  it  is  possible  that  some  light  may  be  incidentally 
thrown  upon  others.  At  least  we  notice  that  such  a  function  as 
7"-|-5v/v — 5,   is,   in  some  respects,  the  same  as  x^+5.r — 5. 

7.  Such  expressions  3.5,  fiinction  of  x,  function  of  a,  function 
of  x-\-h,  etc.,  are  abreviated  into  F{x),  F{a),  F(x-{-h),  or  <f{x), 
f(a),   (f{x-\-h)  or  a  similar  expression.     It  must  be  kept  well  in 


lo  ADVANCED     ALGEBRA. 

mind  that /%  9',  etc.,  are  not  coefficients.  The  expression /(:r) 
will  be  exclusively  used  for  a  rational  integral  function  of  .i",  with 
real  coefficients. 

If  fix)  and /(a),  or  F  (yX)  and  F{a),  occur  together  in  the 
same  discussion,  f{a)  stands  for  what  /(.r)  becomes  when  a  is 
put  for  .r.  Thus,  if  /(.v)  is  x'' — 7.r'+2x-|-4,  then /(«■)  is 
d — 7a-'+2fl'  +  4.  In  the  same  way,  /(.i-+/>)  stands  for  what 
f{x)   becomes,   when    {x-\-h)   is  put  for  x. 

8.  It  is  evident  that 
a^x"-j-a^x"~^-\-a.^x"~'^-\-a.^x''~^-\-  .  .  .  -\-a^^_.^j(r-\-a^^_^x^a^^ 

stands  for  any  rational  integral  function  of  -v,  where  ra  is  a  whole 
number,  and  the  symbols,  a^,  a^,  «^,  a.^,  etc.,  stand  for  any  coeffi- 
cients, positive  or  negative,  integral  or  fractional,  commensurable 
or  incommensurable.  We  confine  ourselves,  for  our  purposes,  to 
?ral  coefficients. 

If  we  suppose/(x)  to  be  divided  through  by  the  coefficient  of 
the  highest  power  of  .v,  then  the  following  will  represent  any /(a): 

.V"+/.,.v"-'+A-i:"-^+A-V"-'+    .    .   .    -f-A-r^"""'"'+A-rl"'^'+A- 

If  none  of  the  above  coefficients  are  zero,  the  function  is  said 
to  be  complete.     The  term,  a,,,  or  A^.  is  called  the  absolute  term. 

9.  Definition.  The  Degree  of  /(.v)  is  the  exponent  of  the 
highest  power  of  x.  In  general,  the  degree  of  a  function  is 
spoken  of  in  the  same  way  as  the  degree  of  an  equation,  and  is 
determined  in  like  manner. 

10.  Definition.  An}^  real  quantit}^  or  any  imaginary, 
which  substituted  for  .i"  in /(.a-)  makes/(.t-)  vanish,  we  will  call  a 
Root  of/(.v}. 

1 1 .  Definition.  A  Rational  Integral  Equation  containing 
one  unknown  is  one  Vv'hich  can  be  placed  in  the  form  /(.r)=o; 
that  is,  in  the  form 

a^"-\-a^x''^^-\-a.pc"~''--\-a.p:"'''^-\-  .  .  .  -\-a^^_^x-\-a^=o       (i) 
A  rational  integral  equation  may  also  be  represented  by 

.r"+/),.v—+A.-^"~"+A^'""H  •  •  •  ^-/'„-r^'+A=o    (2) 

since  equation  (i;  is  unchanged  if  we  divide  through  by  the  co- 
efficient of  .r".  When  the  equation  /(-f)=o  is  written  out  in 
either  of  above  forms  it  is  commonly  spoken  of  as  the  General 
Equation  of  the  «th  degree. 


RATIONAL     INTEGRAL     FUNCTION     OF    .r.  ii 

Matty  equations  which  are  not  in  above  form,  that  is.  are  not 
rational  and  integral,  may  take  that  form  through  the  ordinary- 
processes  of  transformation,  such  as: 

I  x'  -  2 


A" +2       \/ 20— 3-1- 
y/io — 3-r=  X* — 4 
20  — 3.v=.i-' — 8-V-+  16 
.1-^  — 8.v'4-3-v— 4=0 
which  is  a  rational  integral  equation  of  the  fourth  degree.     Some 
equations,  however,  cannot  be  reduced  to  a  rational  integral  equa- 
tion of  finite  degree  by  an}'  process  of  transformation;  such  are 

I  I 

log  x-\-x'-^a;     x-\-c^'=a;   {x-\-a)'^-\-{x-^b)"'^=(  ;  etc. 

12.  Dep'inition.  a  Root  of  an  equation  is  any  value  of  the 
unknown  which  satisfies  the  equation. 

13.  When  the  final  result  in  any  investigation  in  applied  or 
pure  mathematics  takes  the  form  of  a  rational  integral  equation, 
the  problem  of  finding  a  value  of  .v  naturalh'  arises,  and  it  is  one 
of  the  objects  of  the  study  of  the  subject  of  the  Theory  of  Equa- 
tions to  determine  such.  In  order,  then,  to  throw  all  the 
light  possible  on  the.se  equations  of  higher  degree,  we  will  begin 
by  taking  up  the  left  member  of  such  equation,  considered  as  the 
function  of  a  variable,  and  endeavor  to  reach  a  fair  conception  of 
the  nature  and  properties  of  such  functions  in  general.  This 
method  of  procedure  in  itself  might  be  a  justification  for  speaking 
both  of  the  roots  of  an  equation  and  the  roots  of  a  function;  but  it 
is  further  observed  that  these  expressions  are  reall}- the  same,  since 
the  root  of/ (a-),  being  defined  as  the  value  of  .r  which  makes/(.r) 
\-anish,   must  also   be  the    value   which   satisfies    the   equation, 

/(.r)=o.  Nevctheless,  when  speaking  of  a  root  of/(.t-),  the 
thoughts  associated  with  the  expression  in  our  mind  are  different 
from  the  considerations  uppermost  when  we  speak  of  a  root  of 
/(x)=o;  the  first  is  the  more  convenient  expression,  when  we  are 
considering  any  function  of  x  without  reference  to  its  history  or 
application;  the  latter  expres.sion  is  u.sed  when  we  are  considering 
an  equation  of  which /(.i)  is  the  left  member;  we  have  in  mind, 
when  using   /"(a),  .v  as  a  variable,  unrestricted  in   value;    but   in 


12  ADVANCED     ALGEBRA. 

/(.v)=o,  .V  appears  under  restrictions  or  conditions  and  is  fixed   in 
value,  although  unknown. 

14.  Theorkm.     The  difference  between  f{x)  and  /{a)  is  divis- 
ible by  -V — '/.. 

We  are  to  prove      ^  =a  quotient  without  a    remainder. 

X 'I- 

Now/(a')  is  rt„.v"  +  ^?,.r"~'  +  <7.,.f"~-+   .  .  .  +a^^_.,x--\-a^^_^.\-+a^^  Art.  8. 
and/('/)  is    a^^'i."  +  a ^'/■"~^  +  a ,"."~- +  .  .  .  +a^^_./r-\-a^^_^"-\-a^^      Art.  7. 
_j\x)— /{'>:)  ^ 

X — '/ 

^n(-^'"  -  "■")  +  a,  (x—-  «"-') + alx"--'  -  o"-')  +  +  a„^lx'-o')  +  a„_,  (x-a) 

X 'I. 

equals  some  quotient  without  a  remainder,  since  difference  of  like 
powers  is  divisible  bv  the  difference  of  the  quantities  them.selves. 

Q.  E.  D. 

15.  Examples.     Divide   each    of  these  functions    b}^  .v — '/. 
until  the  quotient  does  not  contain  .r,  and  notice  the  remainder: 

1.  .V-— 4A-+7.  3.     4-1"'— 3-^'' +5-1'— I- 

2.  x'' — .r-'+2.  4.      bx' — ex'- — dx-^b. 

16.  Theorem.      IVhenf^x)  is  divided   by  x—i,  the  remain- 
der, after  the  quotient  does  not  contain  .v,  is  equal  to  f  {»■). 

Now,  from  Art.  \x,    -^^-''-'^^''^=.some  qotuient 
x — « 

.-.    ■'^l:^=some  quotient  +^^-^-^ 
x — "-  X — '/ 

.•.     /(a)  is  the  remainder.  Q.  E.  D. 

17.  Theorem.  Any  fix)  is  divisible  by  x  minus  a  root. 
Let  the  root  be  '/. 

Now  from  Art.  1 4,-^-^-^^"-^^"^= some    quotient.      But   /"('/)   is 

X-   a 

zero;  (Art.  \o.). 

-'-^=  some  quotient.  n   t?    n 

18.  Theore.m.     Conversely,    If  any  fi^x)  is   divisible   by   x 
7nimis  »■,   <>■  is  root. 

Then/f.i")  is  (.V — «)Q,  which  will  vanish  when  .r=«,    .'•  <i-   is 
a  root.  Q.  E.  D. 


RATIONAL     INTEGRAL     FUNCTION     OF    nc 

19.      A  short  mctJiod  of  dividing-  miy  f  ix)  by  x — a. 

Suppose  the  /(.r)  to  be  a^y''-{-a^x^  +  a.,x''-\-a.^x^  +  a^x+a.^ 

,,       1  ^    a,x'+a,x^-\-a.,x''-\-a.,x'  +  a,x+a- 

IN owlet    -" — — ' — — — ■' — ^—^ — s 

X — a 

Remainder 


X — a 


where  A,  B,  C,  etc..  are  undeterraiiud  coefficients.      Put  R  {or  Re- 
nt i-indcr. 

Then     a^y ' + c?  ,.v'  +  a.,x'  -f-  a.^x"'  +  a  ^x-\-a.^ 

=  (.i-— a)  I  Ax'-\-Bx'^Cx'-VDx+E^^  -\ 

=  Ax'+B—oA ).v'  +  ( C—"B)x''^  D~iC)x'+{E—'iD )  +  R—'-E. 

Equating  coefficients, 

A  =«„  I  f  ^~  ^n  I 

B—aA  =  a^  I  B=aA-{-a^ 

C—aB=a..  \      ^,        .  C  =  aB+a, 

r^       r-       ■  \     Therefore,      {    j^       r~  \ 
D — aC^a.,  f  I  D='i.C-\rCi., 

E—aD=a\  I  E=aD+a^ 

R—aE=a.  I  I  R=aE+a.^ 

Arranging  the  right  hand  column  of  equations  horizontally: 

Coefficients  in  dividend,   <:7„       +^7,        -\-a.,       -\-a.^        -\-a^        +<^5 

-\-aA       +aB       +aC       +"D      ^-aE 


Coefficients  in  quotient,   A  B  C  D  E  R 

The  following  is  seen  to  be  the  law  of  the  coefficients,  A,  B, 
C,  etc.,  in  the  quotient:  The  first  coefficient  in  the  quotient  is  the 
same  as  the  first  coefficient  in  the  dividend.  The  second  coeffi- 
cient in  the  quotient  is  the  second  coefficient  in  the  dividend  plus 
a  times  A,  the  one  just  found.  The  third  coefficient  in  the  quo- 
tient is  the  third  coefficient  in  the  dividend  plus  a  times  B,  the 
one  just  found.  And  ^;n' coefficient  in  the  quotient  is  the  corre- 
sponding coefficient  in  the  dividend  plus  «  times  the  preceding  one 
in  the  quotient. 

The  process  of  finding  the  coefficients  in  the  quotient,  and  the 
remainder,  is  more  apparent  in  a  particular  case: 


14  ADVANCED     ALGEBRA. 

Find  the  quotient  of  7.1-^+8.1-' — 6a--' — 15.V' — 8.r+4  by  x — 2. 
Coefficients  in  dividend,    7     +8     — 6     — 15     — 8     +4      (2 

14       44         76     122    228 
Coefficients  in  quotient,     7       22       38         61      114    232 

2  T,2 

Henc2 the  quotientis,   7.i-*+22.t''+38-v-  +  6i.i-+ 1 1^-\ — f — 

Find  the  quotient  of  .v^ — 81  by  x — 3: 
Coefficients  in  dividend,  1000     — 81    (3 

3 9^      27 81 

Coefficient  in  quotient,  i         3         9       27  o 

Quotient,  _r'+3A-  +  9.v-f  27,    710  remainder. 

This  method  of  obtaining  the  quotient  is  called  synthetic  divi- 
sion. It  will  evidently  apply  when  the  dividend  is  a  function  of 
any  degree. 

Query  :  What  change  will  there  be  if  the  divisor  is  x-\- '/  ? 

20.  Examples: 

1.  Divide  x*— 5a-'+i2a-+4-1'— 8  by  .r— 2. 

2.  Divide  x^-\- 1  i.v-+36.r+  15  by  A-+5. 

3.  Prove  3  is  a  root  of  x' — 6A-+ii.r — 6.  Art.   17. 
/.  Prove  6  is  a  root  of  x'— i2A''+47.r"' — 72.1-+36. 

5.  Prove  — 6  is  a  root  of -v' — 4.1-' — 29.^-+  i56.r — 180. 

6.  Find  value  of  x'-\-  i6.r — 4.1- -j- 1  when  x^z.  Art  16. 

7.  Find  value  of  .i''+9.i- — 19X— 76  when  .1  =  5. 

21.  Theorem,  a.ssumed.     Every  f{x)  has  at  least  one  root. 
This  means  that  there  is  some  real  quantity,  or  some  imagi- 
nary, which  put  for  x  in 

a^'-\-a^x"~^-^a.^x^'-''-k-  .  .  .  -f  rt,,_,.r+i7„_,.r-f «, 
makes  the  aggregate  value  o.     The  theorem  here  assumed  is   one 
whose  demon.stration  is  too  difficult  to  be  given  in  this  course.      It 
may  be  found  in  Todhunter's  and    other    treatises   on     Theory    cf 
Equations. 

22.  Theorem.      Imaginary  roots  enter  in  pairs. 

Suppose  r+'^^  is  an  imaginary  root  of /(.1-).  We'  will  prove 
that    /'—''/  is    also  a  root. 

We  know /(.v)  to  be  divisible  by  x—y — "'/  (Art  17).  If  it 
is  also  divisible  by  x— /-+''/,  then  ;- — 'V  is  a  root  (Art.  18)  Let 
us  divide /(a-)  by  (.v— /—'>/)  Cr—r+'V),  or  [(.v— ;')"+''']■      Let  the 


RATIONAL     INTEGRAL     FUNCTION     OF     JC-  i5 

quotient  l»e  O  and  stop  when  the  remainder  is  of  the  form  S.v-\-  T. 
Then  /(.r)  is  Qlix-rf  +  '^'I^Sx^  T. 

Put  .1=/'  +  ''/ 

Then,  by  hypothesis,    f{j-\-oi)=o. 

Sr-\-S<H-\-  T=-o. 
.'.     5"'5z=o  and  5=0.          ..      7^=0 
since  the  real  and  imaginary  parts  are  separately  equal  to  zero. 

f(x)  is  divisible  by  (x—-i'  —  ^i){x — y  +  i^i)  without  remainder. 
y  —  in   is  a  root. 

Q.  E.   D. 

23.  An  exactly  similar  proof  shows  that  surd  roots,  of  the 
form  r+\/'',  enter  in  pairs,  if  all  the  coefficients  of/(-v)  are  rational. 

24.  CoROLivARY.  Every  fi^x)  is  -divisible  by  a  rational  inte- 
iiral  fitndio)!  of  the  first  or  second  degree,  since  it  has  one  real  or 
two  imaginary  roots. 

25.  Theorem.  Any  f(yX)  of  the  nth  degree  has  n  roots  and 
no  more. 

a„-v"  +  «j.v"~'  +  a„.v"~^H-  .  .  ,  -\-a^^_.^x'--\-a^^_^x-\-a^^  has  one  root,  real 
or  imaginary,  (Art.  21).  Call  it  '/,.  Then/(x)  is  exactly  divisible 
by  .V — «j    (Art.  17).     The  quotient  is  of  the  form 

a^x"-^-\-Bx"-'-+Cx"-''^  .  .  .  +Hx'+Kx+M. 
Then/(.v)  is  {x—a^)(a^y''-'-^Bx"--'+Cx"-'  +  .  .  .-^ Hx'+Kx+M). 
Since  the  second  factor  is  a  rational  integral  function  of  .r,    it  has 
at  least  one  root.     Call  it  '/,.      Then, 
/■(.v)is  (.V— /,)(.v— ■A,)(«„-i-'-^'+^,.i-'-'+t>"-*+  .  .  .  +R\x-^M^) 

Again,  the  third  factor  must  have  a  root,  suppose  a.^,  whence 
fix)  is  (.V— r/,)(A-— aJ(.r-aJ(a„.v"-^'+.9,.r"-'+  C,x"-'-\-  -^K^x+M.^ 
and  so  on.     Whence 

/(.v)  IS  (.V— ^/,)(.v-«,)(.v— «,)(-V--«J     .    .   .   .{x—".)a^^. 
It  is  obvious,  therefore,  that  /'(.v)  will  vanish  when  x  has  any 
©f  the  values  '-(j,  'a„  "■.^,  etc.,  and  for  no  others.  Q.  E.  D. 

In  the  above,  if  any  of  the  supposed  roots  are  imaginary,  divi- 
sion by  X — a  would  render  some  of  the  coefficients  of  the  quotient, 
B,  C  etc.,  imaginary.  This  could  be  avoided,  however;  for  having 
one  imaginary  root  in  hand,  we  know  that  there  is  another,  and 
could    divide   by  a  quadratic   factor,    as     in   Art.    22.       All  the 


i6  ADVANCED     ALGEBRA. 

coefficients   in   the  divisor  being  real    in  this  case,   those  of  the 
quotient  would  be  real  also. 

26.  Corollary.     Ajiv /(x)  may  be  represented  by 
{x~-<i^){,x—i.X^—'^.,){-^-—<i,) {x—'j.Ja^ 

27.  Corollary.  Every /{x)  of  an  odd  degree  has,  at  least, 
one  real  root.  Since  it  has  an  odd  number  of  roots  altogether,  and 
an  even  number  of  imaginary  roots. 

28.  Examples.     Form  the/(A-)  whose  roots  are: 
/•     3.  2,  1.  _         2.     3,  —2,  I. 

3-     3,  2,    W-^^  2—3s/~i.        4.     o,     3,    v/  2  ,  —>/  2- 

29.  Theorem.  Multiplying  or  dividing  J\x)  by  a  constant 
does  not  affect  the  roots. 

f{x)  is  (.V— aj(.v— «,)Gv— '/J U-—'\.K 

Pffx)  is  p{x—a^){x—a^){x—L.^ (-V— '^.)«o 

/^-^is    '  (.V— a.jC.r— 'Aj(-v-'.,j (-V— '^)«o 

All  of  these,  obviously,  vanish  for  the  same  value  of  .v. 

Q.  E.  D. 

30.  Corollary.  Changing  the  signs  of  all  the  terms  o/f{x) 
does  not  affect  the  roots.     This  is  multiplying  by  — i. 

31.  Theorem.  In  any  fix),  the  coefficient  of  the  highest 
power  being  unity,  the  coefficients  of  the  other  powers  are  functions  of 
the  negatives  of  the  loots. 

Since  we  know  -v"+/i-r"~'+/),,i-"-'+A-^"""'++A-2-'^'"'+/'.-r'^'+A 
can  be  represented  by  (jr — '\){x — 'A,)(-V — /,)  ....  (-v — '-cj,  it  is 
evident  that  the  values  of  A.  A' A-  etc.,  can  be  had  in  terms  of  the 
roots  by  forming  the  product  of  the  binomial  factors,  as  in  the 
demonstration   of  the  binomial  theorem.     Such  product  is: 

-^-i—'WV'—'W'^'—'W,  ■  ■  ■  — '^.-/^,-.'^,)-^"+  •  •  +(^'^'^■/^:.  •  •  «J- 

/>,,  or  thefrst  parenthesis,  is  the  sum  of  the  negatives  of  the  roots. 

A,  or  the  second  parenthesis,  is  the  sum  of  the  products  of  the  neg. 
atives  of   the  root  taken  t7i'o  at  a  time. 

A,  or  the  third  parenthesis,  is  the  sum  of  the  products  of  the  nega- 
tives of  the  roots  taken  three  at  a  time  ;  and  so  on. 

A,  or  the  nth  parenthesis,  is  the  product  of  the  negatives  of  all  the 
roots.  Q-  E.  D. 


RATIONAL     INTEGRAL     FUNCTION     OF    x.  i; 

32.  Corollary.  T/ic  roofs  0/  /(x),  the  coefficient  of  the 
highest  power  being  unity,  are  all  factors  of  the  absolute  term. 

2,2,-  Theorem.  If  two  numbers  be  substituted  for  x  in  fix), 
giving  results  with  contrary  signs,  a)i  odd  )iumber  of  real  roots  lie 
betzveen  the  values  substituted. 

Let  'ij,  o._^,  a,^,  ,  .  .  .  't-y  be  all  the  real  roots  of  /(.i"),  arranged 
in  the  order  of  their  magnitude,  '/.,  being  the  greate.st  and  '/,•  the 
least.  Suppose  /"(.v)  to  be  divided  by  each  one  of  the  real  roots 
and  let  ^{x)  be  the  quotient,  which,  of  course,  is  divisible  by  x 
niinus  each  of  the  imaginary  roots.     Then 

/I1-)  is  (.v-/.,)(.r— /.J(.v-'/.3)   ....    {x~u.,)c{x). 
Now,  from  what  is  known  about  imaginary  roots    (x\rt.  22)    <{'yx) 
must  be  of  the  form 

[Ci  -  dY^e'\{x—ff^g'\  etc. 
and,  since  it  is  the  product  of  the  sum  oi squares,  must  be  positive, 
for  all  real  values  of  .r. 

Now  suppose  -v  to  have  a  value  greater  than  <i.^.  Then  all 
the  factors  are  plus  and  hence  f  {x)  is  plus.  Let  .r  decrease  grad- 
ually and  become  less  than  '/,  but  greater  than  'a^;  then  the  first 
factor  is  —  and  all  the  rest  +,  hence /"U")  is  — .  Now  let  x  be- 
come less  than  «.,  but  greater  than  '/,;  then  two  factors  are  —  and 
the  rest +,  hence /\.v)  is  +.  Again,  let  .v  vary  until  it  is  be- 
tween '/.,  and  '/.^;  three  factors  are  —  and  the  remainder  -f,  hence 
f{x)  is  +.      And  so  on. 

Whence  it  is  evident  that  /  (.v)  changes  signs  every  time  we 
1  ass  over  a  real  root.  Hence,  if  two  numbers  should  be  put  for 
-V  in  /"(.v),  giving  results  wilh  contrary  signs,  there  must  be  an 
odd  nu:n!jer  of  real  roots  between  the  two  values  substituted. 

Q.  E.  D. 
34-     Examples.     Show  that 

/.     .r'— 6.r— 13  has  a  root  between  3  and  4. 
2.     .1' —  i2.x'-f  12.V — 3  has  a  root  between  2  and  3. 
35.     Theorem.      Any  fLx),  the  coeff  dent  of  the  highest  pozcer 
of  X  being  unity,  and  }io)ie  of  the   rest    irreducible   fractions,   cannot 
have  a  root  7vJ/icIi  is  an  irreducible  fraction . 

Letsuch  f{x)  be  .r"  +  /;;,.r""'-)-;;/.,.r"~-+.  .  .  +  w  _.x-\-i)i  ,  where 


,S  ADVANCED     ALGEBRA. 

w,,  w,,,  w.,  etc.,  are   not    irreducible   fractions.       If  this  /(.v)  can 

vanish  when  A' is  an  irreducible  fraction,    let    that   fraction   be    ,. 

Then     /     ^     is  r„  +  w,7„:i^ +  ?«._,  ,„_,,+  .  .  .  +/''/„^,  ,^-)-w,,. 

If  the  value  of  this  is  zero,  it  will  be  zero  when  multiplied  by  ,5""'. 

But   \_^+n!^u"~^  +  )}i J "■"'-.  .  .  +  w„_i,^"~''^  +  ,^"~'w„  cannot  be  zero;  for 

'/"  'J- 

^j  is  an  irreducible  fraction,  since  ,  is;  and  the  rest  of  the  function 

does  not  contain  a  fraction,  since  none  among  «,  r^,  in^,  iii.,,  di..,  etc., 
are  such;  and  a  fraction  must  be  combined  with  a  fraction  to  make 

zero.     Therefore  it  is  impossible  for  ,^  to  be  a  root. 

QE.D. 
36.     Theorem.     Any  J\x)    cati   be    trausfonned  into  a   v'(j') 
whose  coefficient  of  the  highest  power  is  unity  and  ?ione  of  the  other 
coefficients  irreducibk  fractions. 

Suppo.se /(-v)  in  the  form  (Art.   8), 

.v"-f-/',-v"-'+A-^-"-'+  .     .  +/'„-,,-r+A-r^'+A- 
If  any  of  the  coefficients, />,,  p.,,  />,,<   ^^c,   are    fractions,   let    their 
common  denominator  be  q.      Put  .v  =='  ,  then  f{.x)  becomes 

q"     9"        9"  '  9 

Multiply  through  by  q'\ 

y"-VqP,y"~'^fP._y'~''^  ■  ■  ■  +/'~'A-,j'+/A 

where  none  of  the  coefficients  can  be    irreducible    fractions,    since 
each  is  multiplied  by  q,  their  common  denominator. 

Q.E.  D. 

37.  Query.  What  relation  exists  above  between  the  roots 
of/(.r)  and  <s(y)  ? 

38.  Examples.  Transform  these  functions  into  others,  so 
that  the  coefficient  of  the  highest  power  will  be  unit}'  and  no  coef- 
ficients irreducible  fractions: 


RATIONAL     INTEGRAL     FUNCTION     OF    X.  19 

s  ,  14     .       28  8 

9  27         27 

^-      3^-* — 40-v'+ 1 2,ox' — 1 20.V+  27  ■ 
,     II     .,  ,  I 

^-     8  i.r'— 1 8.1-^-36.1-+ 8. 
39.     Theorem.     A?iy /{x)   can    be   transformed  into  a  <s{x) 
-whose  roots  differ  from  those  qff(x)  by  any  assigned  quantity. 

Take/Gv)  as  x-'+p^x"~'p.X'''+  .  .  .  +/„-,-r+A- 
Suppose  .V  to  bej'+//.     Then/(.r)  is /(_>'+//),  or 

{y+h)"+p^{v+hy'-'+p.Av+h)"-'+p.^y+h)"-'+  .  .  .  +A- 
Expanding  the  terms  by  binomial  theorem,   f{x+h)^= 
,,..+  „/„.^-.+  "J»T_il ,.,,-.+       n(n-2}ip),,y..,^ 

P,r-'+  p'^-n^r-'+p!"-^^'  >.y-+  .... 

p.,v"-"--{-pIn  —  2)hv"-''+   .... 
Aj'"-''+  •  •  •  • 
Collecting  in  terms  of  powers  of  r,  we  have/(  ]'+//)  = 

_,''■  +  (;?//+/.,  )!■"-'+   I    "^"^^^^''  +P,in—i)h+p.^^yv"--' 

I  I-2-3  1-2  ^-  '  '  ]- 

which  may  be  called  f  ( )')•  More  terms  can  evidently  be  obtained 
by  expanding  further  by  binomial  theorem.  Since  x^=y-\-h,  y  is 
always  //  less  than  .v,  so  the  roots  of  t-'(j')  must  be  h  less  than 
those  of /"(.t).  By  originally  taking  .\-=i'+//,  the  roots  of  <f{y) 
would  be  //  greater  than  those  of  /(.v).  Q-  E    D. 

40.      HoRNERS'    Method  of   computing  the  coefficients  of 
c-(  )'),   in   above. 

Suppose  f{x)  is  <i:„.v'-f-rz,.r"~'  +  fl!._,.v"~-  +  .  . -{- a ^^_.,x' -\- a ^^_^x + a ^^. 
Put  .1  =  1'+/^  and  suppose/ (-1")  then  becomes 

A,y"  +  A^y"-'+A.j'"--'+.  .  .+.-i,,_,_,j'"---'+.^„_,j'+.^,„ 
where  A^^,A^,A.^.  etc.,  are  undetermined    coefficients.     Now, 
y=x  —  h.     Therefore, 
a^y-"-{-a^x"~^-\-a.^x"~--j-  .  .  .  +«„-.,- v'  +  «„_,.v+(r,, 

=A^iv--hr+A,(x—hy-'+A.sx-hr-'+  ..." 

+  A  „_,(.v— //)■-  -f  A„_,  (x—h )  +  ^„ . 


20  ADVANCED     ALGEBRA. 

Hence,  we  see  that  .-J,,,  the  last  undetermined  coefficient,  is 
the  remainder  when /(a-)  is  divided  by  x — /;.     The  quotient  is 

whence  it  is  apparent  that  ^„_i,  the  next  to  the  last  undetermined 
coefficient,  is  the  remainder  when  this  last  expression  is  divided 
by  x—h;  and  the  same  way  for  the  other  A' s.  ilencc,  the  unde- 
termined coefficients,  beginning  at  the  last,  are  the  successive 
remainders,  as/(-r)  is  continuously  divided  by  x — h. 

As  an  application  of  this,  suppose  it  was  required  to  reduce 
the  roots  of.i' — 5.1" — 5-r  +25  by  2.  We  would  divide  successive- 
ly by  x — 2  by  synthetic  division  as  follows: 

I   c^  ^   +25   2  oi"'  abreviated  thus: 

2  — 6  — 22  I 


I 

3 

—  II 

3 

2 

—  2 

II 

X 

—  I 

-13 

'-^:; 

2 

II 

■13 


I  I       A., 

II  II 

Result:  .r-f .r— 13.1 +3. 

41.  Examples.      Transform  each  J\x)    into  a   cr(  rj    whose 
roots  shall    diffi^r  as  assigned. 

/.     -X"*-|-4.v" — 2.1"'' — 12.V+9,  roots  to  be  3  greater. 

2.  x'' — 9.v''+23.f — 15,  roots  to  be  i  less. 

3.  -v^ — 6.1'— 13,  roots  to  be  3  less. 

42.  Theorem.      Any  f{x)  can   be  transformed  into  a  e(  i-), 
witl:  the  yiext  highest  po'wer  of  v  zvanting. 

In  Article  39    put  .v=i — ^'  ,-   i    e.,  Ii=—      .     Then  the  co- 
;/  n 

efficient  of  the  next  highest  power  of  r  becomes  ( — «— -f /',)oro. 

Q.   E.   D. 

43.  Examples.      Transform  each   into  cr  { y)  with  the  next 
highest  power  wanting. 

/.     .r'— -9.r+23.r — 15. 


RATIONAL     INTEGRAL     FUNCTION     OF    x.  21 

J.     6.x-''  —  I  ijr'^+ 6x —  I . 

/.  Transform  ax^'+dx'  +  cx+d,  where  a,  d,  c,  d,  are  whole 
numbers,  into  a  function  of  the  form  2^-\-h-i-f/2,  where  /  and  m  are 
whole  numbers. 

44.  Theorem.  //  the  signs  of  all  terms  containing  the  odd 
pozvers,  or  the  even  pozvers,  of  x  be  changed,  f{x)  is  transformed  into 
f(-x)  or-f(-x),  the  roots  of  either  of  the  latter  being  the  negatives  of 
the  roots  off(x). 

=  {x-a^){x-a^){x-a^   .   .   .   {x-u.J  (l). 

Putting  — .r  for  x  throughout  the  equation,  we  shall  have/(jr) 
with  the  signs  of  the  odd  powers  changed,  equal  to  the  product  of 
n  factors  of  the  form  ( — x — '/},  or 

fi  _  ^-w  ^  •^'"  -/'r^'"~'+A  v""'-/',.i-"-'+ .  .-/>„_,.r+A'  if '^  '"^^  e^*^^  X 
'.  --i""+/>i-v"~'-/),.r"~--f/',a"~'- .  --/^-r^'+A'  if  ^^  i^  °^^  ^ 
=  (-i)"(.v+'/,)(.v+'/..;(x+'g  .  .  ."U-+'/).  (2). 

Multiplying  this  by  —  i ,  we  will  have  f{x)  with  the  signs  of  the 
even  powers  changed,  or 


-/■(- 


_  (  -  x"+p^x"-  ^-p.,x"  '+pyX"  '•'-..  +p„-r'^'-p„  \in  is  even  ( 
(     x"  - p^x"~^  +Pt'^'"'''- Pt^"~'''  +  •  •  -'rpn-v^  ~J'n  if  '^  is  odd    i 
=  (— i)"^'Cr+r/,)(x+'/.,)(x+'/,)  .  .  .   Cv+'.J  (3). 

Now,  by  changing  the  odd  powers  in  fix)  we  must  obtain 
what  is  contained  in  brackets  in  (2),  which  equals  f{ — x);  and 
from  the  latter  part  of  (2),  it  is  evident  the  roots  are  the  negatives 
of  the  roots  of  (i )  or/(x). 

By  changing  the  even  powers  in/'(x)  we  must  obtain  what  is 
in  brackets  in  (3),  which  equals  -f(-x);  and  from  the  latter  mem- 
ber of  the  equation,  it  appears  that  the  roots  are  the  negatives  of 
those  of  (i)  orf(x).  Q.  E.  D. 

45.  Theorem.  fff(x)  has  r  roots  equal  to  '/.,  the  derivative 
has  r  —  i  roots  equal  to  a,  the  second  derivative  has  r — 2  roots  equal 
to  <>.,  and  so  on. 

Suppose  f{x)  has  r  roots  equal  to  «,  then 

/(^)  =  (x-a)'-^(^) 


22  ADVANCED     ALGEBRA. 

where         <fiX)  stands  for  the  quotient  of/(.v)  by  {x — '/)'' 

Since  Z>.r/(r)  contains  .r-'/.  as  a  factor    r- i  times,    it  has    r— i 

roots  equal  to  '/•. 

Call  the  above  derivative /(.r).      Then,  if  <f,(x)   represents 
the  quotient  as  above, 

/^(x)=(x—ay-W,(x) 

Now  the  derivative  of  this  will  have  r-  2  roots   equal    to    '/;   and 
so  on.  Q-  E.  D. 

46.  Corollary.     {/'/{■^')  has  r  eqtial  roots,  it   and  its  first 
derivative  tvill  have  a  co})imo7i  divisor  of  the  form,  {x  —  ».)'"'^ 

47.  Examples.      Test  each    for   equal  roots  by  finding  the 
highest  common  divisor  of/(-v)  and  its  derivative. 

/.  .;t--'-6.r^+ii.r-6. 

2.  X* '-  5. V '■  +  1 2x' + 4-;v  —  8 . 

J.  2.r'-7.r'+4-^-|-4- 

4  .v'+4-r' -  2.1-' -  1 2.r+9. 

48.  Theorem.     No  f{x)  can  have  more  positive  roots  than   it 
has  changes  of  signs  from  -\-  to  —  and  —    to  -h- 

Suppo.se  we  have  any  polynomial  which,  when  arranged  ac- 
cording to  the  descending  powers  of  .r,  has  the  signs  -|-  and  - 
occurring  in  the  order  given  in  Af  below.  Let  this  polynomial  be 
multiplied  by  .r-'/,  corresponding  to  the  introduction  of  a  positive 
root  '/..  Representing  only  the  signs  in  the  multiplication,  it  is  as 
follows: 
M.      +'      +■'     -\-'     —'     —'      -'     +'     -f'      -"     +'"     -f" 


a. 
b. 

_i 

2 

^3 

+  * 

+^ 

7 

9 

+;; 

u 

P. 

4 

f 

f« 

+' 

r 

— ' 

+'" 

r 

12 

We  have  written  the  signs  of  some  of  the  terms  f  because  a 
+  term  is  combined  to  a  -  term  and  it  is  unknown  which  is  the 
greater.  The  original  polynomial  will  be  spoken  of  as  J/,  the 
first  and  second  partial  products  as  a  and  b,  and  the 
final  product  as  P.  Figures  have  been  attached  to  the 
signs  so  that  they  may  be  .spoken  of  by  number.  It  will 
be     noticed      that     while      we      use     the    particular      order    of 


RATIONAL     INTEGRAL     FUNCTION     OF     x.  23 

signs  given  in  J/,  yet  the  demonstration  we  give  is  perfectly  gen- 
eral,— applicable  whatever   the   order  of  signs  in  the  polynomial. 

(  I ).  We  will  first  sho.v  that  the  product  has  at  least  as  many 
changes  of  sign  as  the  multiplicand. 

The  si.i;ns  in  a  are  the  same  as  those  in  ;J/,  and  the  signs  of  b 
are  those  of  J/ reversed  and  put  one  place  to  the  right. 

Consic.er  an}-  two  consecutive  signs  of  J/,  the  {k — i  )th  and 
X'th.     They  are  either  alike  or  different. 

First  suppose  them  aliker'^  then  the  /'th  sign  of  a  is  the  san-e 
as  the  X'th  sign  of  J/,  while  immediately  under  this  is  the  (y^— i)th 
sign  of  b,  which  is  of  the  opposite  kind,  so  that  the  /tth  sign 
of  the  product /^  is  .-^  Second  suppose  the  {k — i)th  and  X'th 
signs  of  yl/ different ;t  then,  as  before,  the  A'th  sign  of  <a;  is  the  same 
as  the  /'th  sign  of  J/,  but  the  sign  immediately  under  this  is 
the  [k^i  )th  sign  of  b,  which  is  the  {k — i)th  sign  of  J/ reversed,  and 
hence  is  the  same  as  the  /tth  sign  of  a.  Therefore  the  k'&i  sign  of 
P\s  the  same  as  the  /^th  sign  of  M,  unambiguous  and  unchanged. 
Passing  along  the  signs  of  J/  and  /"from  left  to  right,  it  is  evi- 
dent that  \\\^  first  sign  of  P  is  the  same  as  the  first  sign  of  M  and 
that  the  following  signs  of  /^are  all  f,  but  as  soon  as  the  sign  in  M 
changes  from  +  to  —  or  —  to  -f ,  the  second  sign  of  this  change 
appears  in  P  in  the  same  position  as  in  J/.  Therefore,  in  this 
portion  of  M  and  P,  f.ie  signs  begin  alike  and  end  alike,  and  as 
there  is  one  change  of  sign  in  M  there  must  be  one  or  more  (in- 
deed, some  odd  number)  changes  in  P,  and,  as  this  holds  good 
every  time  the  sign  of  M  changes,  therefore  there  are  certainly  as 
many  changes  of  sign  in  /'as  m  M. 

(2)  We  will  show  next  that  there  cannot  be  the  same  num- 
ber of  changes  of  signs  in  P  as  in  M. 

If  tlie  first  sign  of  a  polynomial  is  like  the  last,  the  number  of 
changes  of  signs  between  the  first  sign  and  the  last  must  be  even  ; 
and  it  the  first  sign  differs  from  the  last  sign,  then  there  are  an 
odd  number  of  changes  of  signs  within  it.  Now  the  last  sign  of 
P  is  necessarily  different  from  the  last  sign  of  M.  Therefore  if 
the  first  and  last  signs  of  M  are  alike,  the  first  and  last  signs  of 
P  are  unlike;  and  if  the  first  and  last  signs  of  J/ are  unlike,  the 
first    and  last  signs  of  P  are   alike.      Hence,  if  there  is  an  even 

*  In  this  case  k  would  be  either  2,  3,  5,  6,  8,  or  11,  if  illustration  on  opposite  page  is  used. 
t  To  illustrate  this  reasoning,  k  may  be  taken  as  either  4,  7,  9  or  10  in  work  on  op.  page. 


24  ADVANCED     ALGEBRA. 

number  of  changes  in  M,  there  is  an  odd  number  in  P,  and  if 
an  odd  nnvcvh^x  in  M,  then  an  even  number  in  P.  Hence  /'cannot 
have  the  same  number  of  changes  as  M. 

Now  we  have  shown:  (i.)  that  P  has  at  least  as  many, 
changes  as  J/;  (2.)  that  P  cannot  have  the  same  number  as  J/. 
Therefore  P  has  more  changes  of  signs  than  M.  That  is,  the  re- 
sult after  muhiplying  by  x  — «,  or  introducing  a  positive  root, 
contains  more  changes  of  signs  than  the  original  polynomial. 
Since  at  least  one  additional  change  is  brought  in  for  each  positive 
root  which  may  be  introduced,  no  f{x)  can  have  more  positive 
roots  than  it  has  changes  of  signs  from  -f  to  —  and  —  to  +. 

Q.  E.  D. 

49.  Corollary.  Nof{x)  can  have  more  negative  roots  than 
there  are  changes  of  sigtis  from  -^  to  ~  and  —  to  -j- ,  after  the  signs 
of  all  the  odd  or  even  powers  have  been  changed.      See  Art.  44. 

The  above  theorem  and  corollary  con.stitute  what  is  known 
as  Descartes''  Rule  of  Signs. 

50.  Examples. 

/.  Show  that  x" — i  has  one  positive  root  and  no  other  real 
root. 

This  function,  being  the  difference  of  like  powers,  is  divis- 
ible by  X —  I ;  whence  + 1  is  a  root.  There  is  onl}-  one  change  of 
sign,  hence  it  can  have  no  more  than  one  positive  root,  hence 
none  other  than  +  i .  Changing  the  signs  of  all  the  odd  or  even 
powers  of  x,  there  are  no  changes  of  signs,  hence  no  negative 
roots. 

2.  Show  that  A"'' — I  has  two  real  roots  o\\\y ,  one  +  and  one — . 

3.  Show  that  x^-\-a  has  no  real  roots. 

4.  Discuss  the  roots  x" — a  when  n  is  odd  and  also  when  n 
is  even. 

J".     Discuss  the  roots  of  .i""4-a,  when  71  is  either  odd    or   even. 


CHAPTER  III. 

SOLUTION  OF  NUMERICAL  EQUATIONS. 

1.  As  was  noted  at  the  beginning  of  the  last  chapter,  one  of 
the  objects  of  the  study  of  the  Theory  of  Equations  is  to  establish 
methods  for  solving  any  rational  integral  equation  containing  one 
unknown.  When,  by  any  mathematical  process,  we  obtain  the 
values  of  x  from  an  equation,  the  result  is,  "  by  the  adoption 
of  a  t^rm  primarily  applicable  to  the  mode  or  process  of  its  dis- 
covery, called  the  solution  of  the  equation."  Thus  2  and  3,  the 
values  of  . I  from  .r'— 5,r  +  6=o,  may  be  spoken  of  as  ihe  sohiiion  of 
this  quadratic,  as  well  as  the  process  of  completing  the  square, 
etc  ,  by  which  they  are  obtained.  This  double  use  of  the  word 
solution,  as  designating  either  a  result  or  the  process  by  which  the 
result   is  obtained  will  hardly  lead  to  any  confusion. 

2.  Definition.  A  Solution  ol  an  equation  is  any  function 
of  the  known  quantities  which  represents  the  values  of  x. 

The  solutions  of  the  equations  of  the  first  degree,  x-\-a=o  and 

x'^-\-ax-\-b=o,  are  already  familiar  to  us;  they  are,  respectively, 

J  — azt^/a''' — 4<!>. 

j»r= — a,     and       x= ^ 

2 

Similar  solutions  for  the  equations  of  the  third  and  fourth 
degree,  x'-\-ax''-{-bx+c=o  and  x'-\-ax'-\-bx''-ircx+d=o,  have  been 
obtained,  but  they  are  scarcely  of  value  in  the  solution  of  numer- 
ical equations.  These  are  called  Algebraic  Solutions,  since  they 
involve  only  the  ordinary  algebraic  operations,  such  as  addition, 
subtraction,  multiplication,  division  and  involution  and  evolution 
to  commensurable  powers  and  roots.  Abel,  a  Norwegian  mathe- 
matician, has  proved  that  the  algebraic  solution  of  the  general 
equation  of  fifth  and  higher  degrees  can  not  be  obtained;  that  is, 
the  result  can  not  be  expressed  by  means  of  the  ordinary  algebraic 
symbols  of  operation  alone.  The  function  of  the  known  quantities 
which  expresses  the  values  of  .i",  is  not  within  the  range  of  alge- 
braic analvsis. 


26  ADVANCED     ALGEBRA. 

While  it  is  true  that  an  algebraic  solution  of  the  general  equa- 
tions of  high  degree  is  impossible,  j^et  our  knowledge  of  mathe- 
matical processes  is  sufficient  to  enable  us  to  find  the  real  roots  of 
any  numerical  equation  whatsoever.  The  means  adopted  for 
accomplishing  this  leads  to  a  solution  satisfactory  in  all  respects, 
giving  the  values  of  the  roots  exactly,  if  commensurable,  or  to 
any  desired  degree  of  approximation,  if  incommensurable.  It 
will  be  found  that  the  mode  of  procedure  depends  upon  the 
properties  of  f{x)  established  in  the  last  chapter.  We  shall 
begin  with  an  illustration  of  tlie  general  method  by 
particular  example  and  .shall  afterwards  summarize  the  process  in 
a  general  statement  of  advice  for  any  case. 

3.     Solution  of  the  equation 

x'—\yix'—/^^—iy2x--\-^,y2x-\-  =0.  (I). 

a.  Put.r=V2j',  which  transforms  it  into  the  following  equa- 
tion (II.  Art.  36),  which  has  no  root  that  is  an  irreducible  frac- 
tion: (II.  Art.  35). 

v'C  j')=y— 3/— i6y-f  281-+ 72j'+ 32=0.      ( 2). 

The  real  roots  of  this  must  be  either  integers  or  incommensur- 
able fractions. 

b.  In  seeking  for  integral  roots,  we  need  only  search  among 
the  factors  of  the  absolute  term,  (II.  Art.  32)  which  are:  =ti,  ±2, 
±4,  ±8,  ±16,  ±32.  We  may  now  test  by  dividing  (2)h\y 
minus  each.    (II.  Art.  18). 

I    -3  —16  -(-28  +72  +32   (+1      I  —3  —16  +28  +72+    32  (  +  2 
_j_i   _  2 — 18+10+82  +2    — 2 — 36 — 16+112 


r  —2  —18  +10  +82  +  114  I  —I  —18     —8+56+144 

+  1  not  a  root.  +2  not  a  root. 

I  —3  —16  +28  +72  +32  (  +  3       I  —3  —16  +28  +72  +32  (+4 

+  3         o  —48  —60  +36  +4+4  — 48  — 80  —32 

r       o — 16 — 20+12+78  I  +1  — 12 — 20 —  8         o 

+  3  not  a  root.  +4  w  a  root. 

This  last  quotient  gives  us   an   equation   of  a  lower  degree 
containing  the  remaining  roots:  that  is 

y  +y —  I  iy- — 20) — 8 = o.  (3) . 


SOLUTION     OF     NUMERICAL     EQUATIONS.  27 

We  need  now  tr}'  only  factors  of  8,  of  which  we  have  already 
tried  +1,  +2.  +4.  We  will  try  the  small  negative  numbers  be- 
fore trying  +8.  Dividing  byj'+i,  corresponding  to  a  root— i, 
we  have: 

I  -f  I  —12  —20  +8   (—1 
— I         0+12  — 8 


I       o  — 12  —   8       o  — I  is  a  root. 

The  equation  containing  the  remaining  roots  is 

J''—  12) — 8  =  0.  (4). 

Using  the  untried  factors  of  8,  we  find  no  more  integer  roots. 
Hence  the  equation  must  have  three  incommensurable  roots,  or 
one  incommensurable  and  two  imaginary. 

c.  It  is  now  well  to  test  for  equal  roots,  for  if  two  or  three  of 
these  incommensurable  roots  are  equal,  we  can  find  them  very 
readily.  But  y''—\2y — 8  and  the  derivative  3^ — 12  have  no 
common  di^dsor,  hence  there  are  no  equal  roots.    (II.  Art.  46) 

d.  We  next  attempt  to  locate  the  incommensurable  roots, 
by  assigning  difierent  values  to  j'  and  calculating  the  correspond- 
ing values  of  f(j'j.  (II.  Art.  33).  If  we  put  any  value,  a,  ior y 
in  f(ji'),  we  can  compute  the  value  of  c{a)  by  the  short  method  of 
division;  for  f(a)  is  the  remainder  when  ^iy)  is  divided  hy y — -a. 

(II.  Art.  16). 
) '''  -^-  oy' —  1 2j' — 8  =  0. 

I    +0  — 12      — 8   (  +  2  r    +0   —12    —8   (-f3 


+  2   -f4  —16 

+  3   +9  —9 

I  -f2   —8  —2^=R=^{2) 

I  +3  —3—17  =  ^=^(3) 

I  +0  — 12  —  8  (  +  4 

I   +0  —12  -  8  (-2 

+  4  +16  +16 

— 2  -f  4  +16 

I  +4  +  4  +  8  =  A'=cr(4) 

I   —2  —  8  +8=A'=f( — 2) 

etc. ,      etc. , 

etc. ,      etc. 

28 


ADVANCED     ALGEBRA 


In  like  manner  we  find  the  values  of  <f{r)  when 
values  and  tabulate  them  as  in  the  margin.  The 
first  column  contains  the  values  assigned  to  )'  and 
the  second  contains  the  corresponding  values  of  <f(r)- 
It  is  then  noticed  that  there  is  a  root  between  +3  and 
+  4  and  between  o  and  — i  and  between  — 3  and  — 4. 
(II.  Art.  33). 

e.  The  roots  thus  located  can  be  determined  to 
any  degree  of  accuracy  in  the  manner  following.  Since 
equation  (4)  has  a  root  between  +3  and  +4,  the  first 
figure  of  the  root  is  3.  Then  transform  the  equation, 
by  Horner's  method,  into  one  whose  roots  are  3  less, 
is  as  follows :  i      +  o     — 12     — 8     ( 3 

+  3     +9       —9 
-17 


has 

other 

V 

V'(J') 

—4 

— 21 

—3 

+   I 

— 2 

+  s 

— 1 

4-   3 

0 

—   8 

+  1 

—  19 

+  2 

-24 

+  3 

—  17 

+  4 

+    8 

The 

work 

+  3 


+  6 

+  3 


+  I.S 


I  I  +9 
The  resulting  equation  is 
This 


j'''-|-9_r'+  151/ — 17=0.       ('5"). 
must  have   a  root   between  some  of  these  values:  .0,  .1, 
.2,  .3,  .4,  .5,  .6,  .7,  .8,  .9,  i.o.  By  trial  it  is  found  to   lie   between 
.7  and  .8,  thus: 


I    +9.0  4-15.00   —17.000 
4-   .7   +   6.79  +15-253 


I    4-9-7   +21.79   -    1.747 

The  first  figure  of  this  root  of  (5 
second  ft ^s;  lire  of  a  root  of  {^). 

Now  depress  the  roots  of  ("5)  by 
work  is  as  follows: 

I    +9.0   +15.00  — 17.000  { 

+  • "      +6.79   +15-253 
I    +9.7   +21.79  I  —1.747 


I    +9.0   +15.00  —17.000  (.8 
+    .8   +    7.84   +18.372 

r    +9.8   +22.84  +    1.272 

therefore. 7,  which  is  the 


Ijv  Horner's  method.   The 


1  + 10.4 


+  29.07 


+  11. 1 


SOLUTION     OF     NUMERICAL     EQUATIONS.  29 

The  resulting  equation  is 

j'"'+i  i.ij'"'+29.o7i — 1.747=0  (6). 

This  must  have  a  root  between  some  of  these  values:  .00,  .01, 
.02,  .03,  .04,  .05,  .06,  .07,  .08,  .09.  .10.  By  trial  it  is  found  to 
lie  between  .05,  and  .06,  thus: 

I    -fiiio  +29.0700  — 1.747000  (.05 
+  •05        +-5575   +1-481375 


I    +11. 15   -(-29.6275  —  .265625 

I    +11. 10  -(-29.0700  — 1.747000  (.06 
+  .06       -(-.6696   +1.784376 

I    +11. 16  +29.7396  +   .037376 

Hence  5  is  the  first  figure  of  a  root  of  (6j,  which  is  the  second 
figure  of  a  root  of  (5),  which  is  the  third  figure  of  a  root  of 
(4);    whence  this  root  of  ('4),  correct  to  three  figures,  is  3.75. 

We  now  depress  the  roots    of   (6),  by  .05.     The  work    is    as 
follows: 

1      +11. 10     +29.0700     — 1.747000     (.05 
•05  -5575  I -481375 


1      +11. 15     +29.6275 

^5 -5600 

I      +11.20  I  +30.1875 
•05  I 


^65625 


I  I  +11-25 

The  resulting  equation  is, 

j''+ii.25_i''+3o.i875_r  —  . 265625.  (7). 

By  trial  this  equation  is  found  to  have  a  root  between  .008  and  .009. 
Whence  8  is  the  first  figure  of  a  root  of  (7 ),  the  second  figure 
of  a  root  of  (6),  the  third  figure  of  a  root  of  (5),  or  the  fourth 
figure  of  a  root  of  (4),  whence  a  root  of  (4)  complete  to  four 
figures  is  3.758.  It  is  evident  that  the  root  can  be  determined 
to  2ix\y  degree  of  accuracy  by  a  continuation  of  this  process. 


30  ADVANCED     ALGEBRA. 

In  like  manner  we  might  have  found  the  other  incommen- 
surable roots  of  (4),  either  the  one  between  ^3  and  — 4,  or  o  and 
—  I.  For,  an  incommensurable  negative  root  can  be  found  if  the 
negative  roots  be  transfoimed  into  positive  ones  before  applying 
Harnett's  Method.     This  can  be  done  by  II.  Art.  44. 

We  can  obtain  the  approximate  values  of  the  other  two  roots 
from  the  quadratic  resulting  from  removing  the  approximate  root 
from  (7 ). 

I    +11.250  +30.187500 — .265625000  (.008 
.008  .090064       .242220512 

I    +11.258  +30.277564 

J'-'  +11.258J'  +30.277564=0  (8). 

y-  +11.258J'  +31.695641  =  1.418075 
whence  )'=  — 4.439  and  — 6.S19. 

But,  remembering  that  these  roots  are  3.75  less  than  those  of  (4), 
we  really  have,  as  the  roots  of  (4), 

j'=  — 689  and  — 3  069- 

/.  Finally,  collecting  all  the  values  of  r  found,  and  remember- 
ing that  .v=  Y'zy,  we  obtain  these  results  as  the  solution  of 

y? —  lY-zX^— 4.1'' —  3 1 2  x" + 4  V2  .r  +  I  =  o 

J' =  — 4,  or  — I,  or  +3.758+,  or  — .689,  or  — 3. 069 

x=  — 2,  or — 1-2,   or  +1.879+,  or — .345,  or — 1.535. 

4-  The  Principle  of  Trial  Divisors.  The  process  given 
above  for  finding  the  successive  figures  of  a  root  would  be  found 
to  be  ver>^  tedious,  since  each  successive  figure  is  determined  from 
among  several  digits  by  actual  trial.  But  it  wnll  be  found  that 
after  one  or  two  figures  of  the  root  are  obtained  as  above,  that  a 
suggestion  of  the  next  figure  can  generally  be  obtained  in    a  ver>' 


SOLliTlON     OF     NUMKRICAL     EQUATIONS.  31 

simple    wa}-.       To   illustrate    this,    consider   equation    (7)    in  the 
last  article: 

k''+  1 1. 25)'' 4- 30. 1875J' — .265625=0. 

We  know  that  v  is  some  number  of  thousandths  phis  some- 
thing. We  are  determining  the  figures  of  the  root  one  at  a  time, 
and  at  present  merely  desire  the  number  of  thousandths.  By  prop- 
er transformations  in  the  equation  it  is  evident  that 

^^  -265625 


-f- 1 1.251'+ 30. 1875 

Now,  since  1'  is  known  to  be  a  fraction,  and  less  than  one  hun- 
dredth, the  most  valuable  lermin  the  denominator  of  the  fraction  is 
30.1875;  for,  the  higher  the  powers  of  y  the  less  account  they 
are  when    )'  is  a  fi  action.      Hence 

!■=    ■"   ^:^   ^  nearlv=  .000879-!-  (8). 

30.1875 

Whence  it  is  quite  certain  that  8  is  the  first  figure  of  j',  or 
the  fourth  figure  of  a  root  of  (4).  Therefore,  in  any  case,  when 
two  or  three  figures  of  a  root  have  been  obtained,  a  suggestion  of 
the  next  figure  can  be  had  by  dividing  the  absolute  term  of  the 
depre.s.sed  equation  by  the  coefiicient  of  j/. 

This  is  known  as  the  principle  of  trial  divisors.  Of  course  as 
we  find  more  figures  of  the  root  and  continually  depress  the  roots 
of  the  equation,  the  smaller  r  becomes  and  the  more  surely  can 
we  rely  upon  the  .suggested  value.  Thus  two  figures  of  the  ap- 
proximate value  of  1'  in  (8)  are  really  correct,  as  will  be  .seen. 

We  now  give  a  more  elaborate  arrangement  of  the  work  of 
Horner's  method  as  used  in  the  last  article.  The  lines  that  ex- 
tend across  the  page  are  the  divisions  between  the  successive 
depressions.  Follow  each  line  across  and  the  numbers  beneath 
it  are  the  coefficients  of  the  equation  with  roots  depressed.  The 
fifth  figure  of  the  root  was  obtained  by  the  method  of  trial  divisor, 
i.e.  I)y  dividing    .r)23404488  by  30.367756. 


32 

1 

o 
_3 

3 

3 

6 

3 

9.0 

_-l 

9-7 

•7 

10.4 

ADV.^i 

SIC  ED     ALGKBRA 

— 12 
9 

<2, 

—  8       (3-7587 

—  9 

—  3                                  —17.000 
18                                       15-253 

15.00 
6.79 

21.79 
7.28 

—  1.747000 
I-481375 

—  .265625000 
.242220^12 

29.0700 

•5575 

29.6275 

.  5600 

—     .023404488 

II. 10 

05 

1 1 . 1 5 

•05 
11.20 

•05 

30.187500 
.090064 

30.277564 
.090192 

30.367756 

11.250 

008 

11.258 

.008 
11.266 

.008 

5- 
cedure 

11.274 
The   example  i 
bv    which   we    1 

n 
nai 

Art.  3  illustrates   the  method  of  pro- 
/     determine     the     real    roots  of   any 

numerical  equation.      We  briefly  summarize  it  in   the   following 
statement  of  advice  for  any  case: 

Any  numerical  equation  being  given: 

a.  Transform  the  equation,  if  necessary,  so  that  the  coeffi- 
cient of  the  highest  power  in/ (.rj  is  unity  and  none  of  the  rest 
irreducible  fractions.      Query:  Why? 

b.  Search  among  the  positive  and  negative  factors  of  the 
absolute  term  for  integer  roots,  by  dividingy"(.x ),  or  f  ('.!').  by  x, 
or    r,    minus   each    factor  by  synthetic  division.      Use  the  nunieri- 


SOLUTION     OF     NUMERICAL     EQUATIONvS.  33 

cally    smallest   values    first.      Depress  the   degree  of  the  equation 
whenev'er  a  root  is  found. 

r.  When  all  integer  roots  are  found,  test  for  equal  roots,  by 
noting  whether  the  function  and  the  derivative  have  a  common 
divisor. 

d.  Locate  the  incommensurable  roots  by  II.  Art.  33,  and 
approximate  them  as  desired  by  Horner's  method. 

e.  As  soon  as  a  quadratic  is  obtained,  solve  it  in  the  ordi- 
nary way. 

/".  Remember  that  if  the  number  of  the  roots  found  does  not 
equal  the  degree  of  the  equation,  that  the  rest  may  be  imaginary. 

6.  Examples.     Find  the  roots  of  the  following: 

1.  a' — 9.1:'+ 23.1- — 15=0. 

2.  x*-{-2.\'' — 2  1.1"' — 2  2. r+ 40^0. 
3-  a'— 51-'— i3.v-  +  53.r+6o=o. 
7.  .r'  — 6.i-''+8.v' — 17-1-+  10=0. 

5.  -i-*  +  4.r' — 2.1"'' — i2.v"+9=o. 

6.  2.r' — 7.1-*  — 9.i"'+33-v'+  17-r — 30=0. 

7.  .v' — 3 1 J  .v'-+  2.1-1-  2=0. 

8.  .v' — 3=0.      By  Horner's  Method. 
9-  -v'— 15=0. 

TO,     X' — 3.1-1-1=0. 

/  / .     4.1"^ —  1 3.r^  -f- 1 2.r  —  30= o. 

r2.     8.r — 25.v'-(-  24.1-—  75=0 

7.  ALCiEBRAic  Solution  of  the  Cubic,  generally  known 
as  Cardan's. 

Suppo.se  -r'-|-/>,.r'-f-/>,,i--|-/>3=o  in  the  form 

r'-f  ^7.1-1- /^=o.  II.  Art.  42. 

Let  x=y+2,  then 

That  is,  /-\-2'+{xv"+a)(y+3')  +  b=o. 

Now  we  will  make  another  assumption  with  respect  to  r  and 
z.  namely  that  3i'.e-|-a=o.  We  can  do  this,  since  the  only  re- 
striction placed  upon  the  hvo  unknowns,  y  and  ,?,  is  that  their 
sum  equals  a  root  of  the  equation;  and  upon  two  unknowns  two 
conditions,  not  inconsistent,  can  be  placed. 


34  ADVANCED     ALGEBRA. 

Therefore,  /+z^+b=o. 

A  1  •  ^  •<  (  «  )     '^  , 

And,  since  ^= ,      y+  -, .•   +b=^o. 

Or  f+h^—{yiaf=o. 

Whence,  since  this  is  a  quadratic  m  terms  of y, 

z'=—b—/=  -  1 2 b^V{y2br  +  {haf 
Therefore, 

A-=  {   -  1/2  (5  +  >/(  i^/^)2  +  ,.3«):^ }  'o  +  I  _  I  .  ^— ^[.a)-+  [^af  }  -' 

Here  jir  is  expressed  as  a  function  of  the  known  quantities,  a 
and  b.     See  Art.  2. 

Now,  if  (ysaf  is  negative  and  greater  than  (  ^jl^f,  the  above 
function  becomes  the  sum  of  the  cube  roots  of  two  imaginaries. 
Since  no  arithmetical  or  algebraic  method  exists  for  finding  the 
cube  root  of  imaginaries,  "  the  roots  are  in  this  case  presented  to 
us  in  a  form  which  is  very  inconvenient  for  arithmetical  pur- 
poses." For  this  reason  the  solution  is  of  but  little  practical 
value. 

8.  Algebraic  Solution  of  the  Biquadratic,  generally 
known  as  Descartes'. 

Suppose  -r*+p^x'+p.,x'+p.^x-\-p=o  in  the  form 
x*-\-a  x'-  -hbx-\-r=o. 

Suppose    x'-\-ax'-\-bx+c=(x'  +  Ax  +  B)(x''+  Cx+D) 
where  A,  B,  Cand  D  are  undetermined  coefficients. 

Then      x'  +  ax-  +  bx+c 

=x'+{A  +  C)x'+(B+AC+D)x'  +  {AD+BC)x+BD. 
Equating  coefficients: 

A  +  C=o  whence  A= — C 

B  +AC+D=a  "  B+D—A'=a 

AD-^BC=b  "  A{D-B)  =  b 

BD=c  "  BD=c 

Therefore    D+B=a  +  A';  D—B=  ^   ;  BD=c. 

A 


SOLUTION     OF     NUMERICAL     EQUATIONS.  35 

Finding  B  and  D  from  the  first  two  equations  and  substitut- 
ing in  the  third, 

Multiplying  out,  recognizing  a  product  of  sum  and  difference, 

which  is  a  cubic  in  terms  of  A' .     Therefore  A  can  be    found  and 
then  B,  Cand  D  from  the  equations  above.      Finally,  four  values 
of  X  can  be  found  by  solving  the  two  quadratics, 
.r-  +  .^r+^=o  and  .r+Ci-f /7=o. 


CHAPTER  IV. 


GRAPHIC  REPRESEiNTATION  OF  EQUATIONS. 

r .  The  Axes.  Any  point  in  a  plane  may  be  located  by  a  sys- 
tem of  latitude  and  longitude  ;  that  is,  by  giving  its  distances 
from    two    fixed  or    standard  lines  of  indefinite    length.     These 

Y 


+  3 


X' 


-r=+4 


■  stands  for  the  adsn'ss(7S  or  /on<^tV//(/f: 
stands  for  the  on1i)iates  or  latitudes. 


GRAPHIC     REPRESENTATION     OF  EQUATIONS.  37 

Standard  lines  are  called  the  Axes,  and  are  distinguished  as  the 
.V  axis  and  the  y-ax/s.  In  geography  they  are  not  lines  but 
arcs  of  circles  and  are  known  as  the  Equator  and  Standard 
Meridian.  In  the  figure  A'A''  is  the  .x--axis  and  W  is  the  j'-axis. 
They  are  generally,  though  not  necessarily,  taken  at  right  angles. 
Their  point  of  intersection,    O,  is  called  the  Origin. 

2.  Co-ordinates.  The  distances  of  a  point  from  the  two 
axes  are  called  its  Co-ordinates.  The  co-ordinate  parallel  to  the 
.v-axis  is  called  the  Abscissa  ;  the  co-ordinate  parallel  to  the  j'-axis, 
the  Ordinate.  These  displace  the  words  Longitude  and  Latitude. 
Abscissas  are  always  represented  by  an  x,  and  ordinates  by    a  y. 

3.  Signs.  Abscissas,  when  measured  to  the r?^/;/,  are  reck- 
oned positive,  and  when  to  the  left,  are  negative.  Ordinates,  when 
measured  ///»  are  positive,  and  when  doivn,  are  negative. 

4.  Examples.  Draw  the  axes  and  locate  the  following 
points.  Use  any  convenient  length  as  the  unit  of  measure;  for  ex- 
ample, a  half-inch. 

/.  (.i-=-2,_i'=  +  6)  2.  (.v=  +  i,j=— 3)  3.  (.r=o,j'=-2) 
i.  (.v=  — 4,  )'=  I2)  5.  (-v=-|-5, j'=o)  6.  Locate  the  follow- 
ing points  and  r(V?;/<'YY///<^v;/  /;/  order  zcith  lines:  (-r= — 2,  v=  — 7) 
(.v=-i,j'=-5  )  (.r=o,j'=-3)  (.r=-|-i,j'=  -  I)  (.t-=  +  2,_>'=-f  i) 
'  »  =  +  .v.i'=  +  3)  <-i"=  +  4- J'=  +  5)  (-^"=  +  5.  J=  +  7)- 

5.  When  a  number  of  points  are  to  be  located  by         x  I  y 
their  abscissas  and  ordinates,  these  are  best  given  in  a 
tabular  form  as  in  the  margin;   each  pair  of  values  lo- 
cating a  point.     Those  here  presented  are  the  same  as 
the  ones  in  the  last  example- 

Now,  it  will  be  noticed  that  if  the  different  values 
of  .V  appearing    in  this  table  be  assigned  to  x  in  the 
equation  2X — 3=  r,  that  the  values  of  j  resulting,  will 
be  identical  with  those  tabulated  above.   Hence  we  may 
look  upon  the  above  table,  and  the  points   located    therefrom,  as 
having  come  from  this  equation.     In  the  same  manner,  other  values 
could  be  asigned  to  x    and    the   corresponding    values  of  i' found, 
and  the  talkie  therebv  extended.      Or,    any    number   of   fractional 


2 

—  7 

1 

—5 

0 

—3 

+  1 

—  I 

+  2 

+  1 

+  S 

+  3 

+  4 

+  5 

+  s 

+  7 

38  ADVANCED     ALCxERRA. 

values  of  A"  might  be  interpolated  and  intermediate  values  of  r 
obtained.  In  either  case,  all  the  points  thus  located  by  the  pairs 
of  values  would  be  found  to  lie  along  the  same  line.  For  this 
reason  the  line  is  said  to  be  a  Graphic  Representation  of  the 
Equation  2x — 3=1'. 

The  student  will  now  proceed  to  attempt  the  graphic  represen- 
tation of  each  of  the  following  equations.  Enough  values  of  .v 
and  J' are  to  be  tabulated,  and  enough  points  located,  to  satisfy 
the  student  in  each  instance  of  the  character  of  the  graphic  repre- 
sentation of  the  equation.  Each  equation  should  be  plotted  on 
paper  about  8  X  10  inches,  marked  with  the  equation  and  its  num 
ber  and  kept  for  reference. 

Also 


6.     Equations.     Plot  Nos.  3  and 

4   on  same   a 

Nos.  9  and  10. 

/. 

7-^— 3=.i'-                         /• 

0  X-\-2  =  _V. 

2. 

4.1  +  5=)'.                         8 

—3.1— 4=   I' 

3- 

3-^'— 3=  J'-                         9- 

— 5-1— 4=.i' 

i- 

3.1 +4=  ]'.                         /n. 

— 5.v+3=j' 

5- 

-\- — 3=  1'.                           //. 

12.1-1-4  = 

6. 

'2-V+5=J'.                                 /2. 

ax+b=r. 

7.  Queries.  These  queries  apply  to  the  above  equations 
of  first  degree,  or  to  any  other  equations  of  the  first  degree  in  the 
form  f{x)=-y.  They  are  to  be  answered  by  studying  the  plots. 
Of  course  the  first  conclusions  reached  will  be  merely  probable  in- 
ferences leased  upon  the  consideration  of  a  limited  number  of 
cases;  but  the  student  should  afterwards  endeavor  to  find  the  rea- 
S071  why  his  statement  is  true,  by  which  means  he  will  probably 
detect  a  rigid  proof  for  his  former  inference.  This  will  be  a  valu- 
able course  of  discovery  for  the  student. 

The  numbers  in  parentheses  following  the  queries  refer  to  the 
equations  and  are  intended  to  direct  the  student  to  such  plots  as 
will  suggest  the  answers. 

/.     What  do  equations  of  the  first  degree  represent? 

2.  In  case  the  absolute  terms  of  several  equations  of  the 
finst  degree  are  alike,  how  do  the  ])lots  compare?  {/and  ?). 
(S  and  9),    (./  and  //). 


GRAPHIC     REPRESENTATION     OF    EQUATIONS.  39 

J.      What  does  the  absolute  term  tell  about  the  plot? 
^.      How   do   the   lines  compare  if  the  coefficients   of  x  are 
alike?     (j  and  ^),  ig  and  /o). 

5.  What  effect  has  the  sign  of  the  coefficient  of  -V  upon  the 
direction  of  the  line?  (/  and  S),  (/,  2,  j,  /,  5,  6,  and  S,  p,  /o,  11). 

6.  What  effect  does  increasing  the  numerical  value  of  the 
coefficient  of  x  have  upon  the  direction  of  the  line?  (5,  J,  i), 
(8,  g),  (7,  6,    5,  /,  ^,  /),  (c?,  g,  11). 

8.  Equations.  Of  course  we  might  tabulate  the  values  of 
.r  and  y  from  any  equation  containing  these  variables,  such  as 
.r'-f-.ri'+r"=i9,  and  attempt  its  graphic  representation.  It  is  our 
present  purpose,  however,  to  confine  our  attention  to  the  repre- 
sentation of  equations  of  the  form  /'(.r)=  r,  for  then  we  will  be  able 
to  discuss  geometrically  many  of  the  properties  of  /(.r)  which 
were  considered  algebraically  in  Chapter  II.  This  is  possible, 
since,  in  such  equations,  y  is  always  the  value  of  /'(.i)  correspond- 
ing to  a  particular  value  of  x,  and,  consequently,  the  ordinates  in 
the  plot  are  graphic  representatives  of  the  different  values  of 
/(-v)  .  As  a  consequence  the  assemblage  of  ordinates  brings  to 
our  e>-e  the  changes  in  value  which  f  {x)  undergoes  when  x 
varies.     This  statement  will  be  better  appreciated  as  we  proceed. 


^3- 

.1"— 2.r  — 3=)'. 

18. 

.V-— 6.v+8=r. 

14. 

.t"-4.r-5=i'. 

ig. 

.r' — 6-V+9=i'. 

rS- 

.r'— 6.v-7=r. 

20. 

.r'-— 6.v+i3=r. 

16. 

x'  —  bx       =r. 

21 . 

x'^-^.x—:^=y. 

17- 

A-^-6.v+5=i'. 

22. 

.i-Hox— 3=1'. 

g.  Queries.  7.  In  what  respect  would  you  say  the  plots  of 
all  these  equations  of  second  degree  are  alike? 

8.  What  relation  is  there  between  the  position  of  the  lowest 
point  of  the  curve  and  the  coefficient  of  .r?     (/j,  21,  22). 

g.  Changing  the  absolute  term  has  what  effect  on  the  plot? 
{is— 20). 

1  D.  Equations-  It  will  be  fouud  in  plotting  equations*  of 
higher  degree  that  many  times  the  value  of  j'  are  very  large  com- 

*The  wjrd  "  equation,"  unless  the  contrary  is  stated,  is  now  being  used   for  the  com- 
plete expression  "  equation  of  the  form  /(.»)  -y." 


40 


ADVANCED     ALGEBRA 


pared  to  the  corresponding  values  of  x.  This  would  require  a 
very  long  piece  of  paper  for  plotting.  Such  a  requirement  can  be 
avoided,  however,  by  using  a  smaller  unit  of  measure  for  the  j/'s 
than  for  the  x's;  for  example,  an  inch  as  the  unit  for  x  and  an 
eighth  of  an  inch  for  j;  or  ten-sixteenths  of  an  inch  for  x  and  one- 
sixteenth  fori';  or  an}^  such  arrangement,  the  student  exercising 
his  judgment  in  each  instance.  By  this  means  the  essential  prop- 
erties of  the  curves  are  retained;  the  effect  is  the  same  as  if  the 
curve  was  plotted  properly  on  a  long  piece  of  paper  and  after- 
wards the  paper  shTiink  a  great  deal  in  the  direction  of  its  length. 


A  PLOT  OF  AN  EQUATION  OK  THIRD  DEGREE- 


23- 

2.r'— 7.v-'  +  4.v-f  4=)'. 

27- 

.v-'+2.i-+3  =  i'. 

2i- 

.r-h3.r-f2.r=i'. 

28 . 

.1-'— 3.1  =  1'. 

25- 

.1-'— 5.v-+i  =  r. 

29. 

x  +  T,x^y. 

26. 

.r'  —  6.v-'+  I  i.r — 6=j'. 

3(^- 

.1'+  2.r— 3.i'— 4.1+  2=0 

GRAPHIC     REPRESENTATION     OF     EQUATIONS.  41 

11.  Definition.  Any  portion  of  a  curve  having  the  same 
direction  as  the  .r-axis  is  called  an  Elbow  of  the  curve. 

12.  Definition.  Any  point  in  a  curve  at  which  the  curv- 
ature changes  from  convexity  to  concavity,  or  vice  versa,  is  called 
a  Point  of  Inflexion.     The  middle  point  of  the  letter  S  is  such. 

13.  Queries.  9  In  the  plot  of  J\x)=y  what  represents 
the  roots  of  /(.v),  or/(.i-)=o? 

10.  How  is  it  shown  that  if  different  values  be  substituted  for 
-x-,  giving  results  with  contrary  signs,  that  at  least  one  real  root 
lies  between  tho.se  values? 

//.  Can  every  curve  of  third  degree  be  divided  into  two 
equal  parts? 

14.  Equations. 

ji.     x' — 4.r'-)-.i-'-j-  7.1- — 3=  r. 

32.  x' — .r — 1 3.v-'+.i -f  1 2=y. 

33.  x' — ^2.v' — 7.1" — S.v-(-  l6=J'. 

34..      2.r' — -x'' — 9.i"  +  33.r-(-  1 7. r— 30=7. 
33.     X' — 3-1-* — 1 6.x -(-48=  I'. 
36.     .r' — 5-1  =  1'. 
3j.     .r'+5.v=i'. 

15.  Queries.  12.  What  is  the  difference  between  the 
cur\'es  of  even  and  odd  degree  ? 

13.  What  shows  that  every  f\x)  of  odd  degree  has  at 
least  one  real  root  ? 

//.  How  many  times  can  a  line  parallel  to  the  r-axis  cross 
a  curve  representing  any/(.v)=j'? 

13.  How  manj'  times  can  a  line  parallel  to  the  .r-axis  cross 
a  curve  of  any  degree  ? 

if).  What  shows  that  imaginary  roots  enter  in  pairs  ?  (Note 
eqs.  15-20  and  query  9.) 

77.  How  are  hvo  equal  roots  indicated  in  the  plot  ?  (19,  23, 
24,  30). 

18.  How  many  elbows  does  a  curve  of  given  degree  con- 
tain ?     (Note  27,  29,  33,  35,  37. ) 

16.  Equations.  In  the  following,  the  derivative  of  the 
J\x)  taken  from  a  preceding  equation  (de.signated  b}^  number)    is 


42  ADVANCED     ALGEBRA. 

first  to  be  found.  This  is  then  to  be  placed  equal  to  r  and  plotted 
on  the  same  axes  as  the  original  curve,  using  also  the  same  unit 
of  measure  as  was  used  before.  When  speaking  of  the  new  plot 
with  reference  to  the  old  one,  it  is  called  the  Derivative  Curve. 
Straight  lines  are  to  be  drawn  parallel  to  the  j'-axis  through  the 
intersection  of  the  derivative  curve  with  the  X-axis.  Also,  simi- 
lar lines  are  to  be  drawn  through  the  elbovy^s  of  the  derivative 
curve. 

j8.     Derivative  Curve  to  17. 

jg.     Derivative  Curve  to  26. 

40.  Derivative  Curve  to  30. 

41.  Derivative  Curve  to  32. 

42.  Derivative  Curve  to  36- 

17.  Queries,  ig-  What  does  the  derivative  curve  tell 
about  the  original  curve  ? 

20.  How  can  you  find  the  number  and  locations  of  the  el- 
bows of  any  curve  ? 

18.  Equations. 

4J.     x' — ]/^x=y.  46.     {X — iY=y. 

44.     x'—.oix=y.  47.     (.r— 1)^=1'. 

/5.     X' — o.v^j'.  48.     (x — I  )•'=_)'. 

19.  Query.  21.  What  peculiarity  of  the  plot  where  we 
have  several  equal  roots  ? 

20.  Equations.  Take/(.r)  from  the  equation  designated, 
transform  it  as  required,  place  it  equal  to  r  and  plot  on  the  .same 
axes  with  the  original. 

4g.  Equation  formed  by  decreasing  roots  of  (26)  by  two. 
^o.  Equation  formed  by  increasing  roots  of  (34J  by  one. 
^i.     Equation  formed  by  increasing  roots  of  (32)  by  three. 

21.  Query.  22.  What  effect  has  increasing  or  decreasing 
the  roots  upon  the  curve  ? 

22.  Equations,  not  in  the  form,  f(x)=y,  which  the  stu- 
dent may  plot  after  tabulating  the  values  of  .v  and  )-. 

a.  x'-\-y'=2=^.  c.      36.a''+iooi''=36oo. 

b.  .V-'— )-=i6.  d.      i'-'  =  8.v. 


CHAPTER  V. 

DETERMINANTS. 

I.  lyCt  US  take  two  simultaneous  equations  of  the  first 
degree  and,  from  tliem,  find  the  vahies  of  jf  andjv. 

a^x+b^y=c^  (i) 

a.A'~\-li,y^c.,  (2) 

Multiph'  ( I )  by  A,  and  (2)  by  /',,   whence 

a^h.,x+b^b.,y^c^h.,  (3) 

a.,b^x+b^b,y=c,b^  (4) 

Subtracting  (4)  from  (3) 

( «|  A, — a.,b^  )x=c^b., — c^b^ 

,      "       "     c,b—c,b,  ^  . 

hence  x=~^ ^  (s) 

a^b.^—a.J)^ 

Now  multiply  (i)  by  a.,  and  (2)  by  a^,  whence 

a^a.,x+a.,b^y=a.f\  (6) 

a^a.y-\-a^b.,y=^a^c.,  (7) 

vSubtracting  (6)  from  (7) 

(«,  A, — a.Jb^ )  )'=  a/; — -a,/, 

,  ax., — a.,c. 

hence   r=     /  (8 ) 

It  is  to  be  noticed  that  the  denominators  in  the  values  of  .v  and 
y  (equations  (5)  and  (8))  are  just  alike  and  contain  only  the  co- 
efficients of  .r  and  I  in  the  given  equations  (ij  and  (2).  This  ex- 
l)ression  a^b.,—a,b^  iscalled  the  Determinant  of  the  four  quantities, 
<?p  a.,,  /?,,  A,,  and  we  agree  to  write  it  in  the  form  of  a  square, 
where  the  numbers  «,,  ^,,  «'.„  A,,  are  arranged  in  the  same  order  as 
they  appear  in  the  equations  ( i )  and  (2)  with  a  vertical  line  before 
and  after  the  square,  thus 

Now  in  order  that  this  shall  mean  the  same  as  ajj., — a.,b^  it  is 
evident  that  we  must  take  the  product  of  the  diagonal  terms  run- 


44  ADVANCED     ALGEBRA. 

ning  downward  to  the  right  and  from  this  product  subtract  the 
product  of  the  diagonal  terms  running  upward  to  the  right. 

Returning  now  to  the  value  of  .r  in  equation  (5)  it  is  seen 
that  the  numerator  ^-jA^—r/^j  is  like  the  denominator  except  that 
Tj  and  c,  are  written  in  place  of  a,  and  a.,,  and  this  expression  is 
called  the  determinant  of  the  four  quantities  r,,  b^,  c,  b.„  and, 
like  the  other,  may  be  written 

U.    K\ 

provided  it  is  agreed  to  interpret  this  square  array  of  numbers  the 
same  as  the  previous  square  array  was  interpreted.  In  the  value 
of  J'  in  equation  (8)  it  is  seen  that  the  numerator  a^c, — a.,c^  is  like 
the  denominator  except  that  c^  and  c.^  take  the  place  of  b^  and  b^, 
and,  just  as  before,  the  expression  a/,  ~^'f\  ^^Y  be  written  in  the 
form  of  a  square,  provided  the  square  array  of  numbers  be  inter- 
preted as  before. 

The  values  of  .t"  and  )'  written  in  the  notation  just  explained 
are 


''", 

b^ 

c, 

b., 

«, 

b, 

a., 

/^„ 

and  r 


I  a,     ^ 

I  '^\      ^\ 
a.,     b.. 


Each  of  these  determinants  in  the  numerators  and  denomina" 
tors  of  the  values  of  x  and  r  is  expanded  (i.  c.  written  out  in 
the  ordinary  form)  by  the  same  rule,  viz. :  Multiply  together  the 
numbers  in  the  diagonal  running  downward  to  the  right  and  from 
this  product  subtract  the  product  of  the  numbers  in  the  diagonal 
running  upward  to  the  right. 

The  student  should  carefully  notice  these  values  of  x  and  y. 
In  each  case  the  denominator  is  the  determinant  obtained  by  writ- 
ing the  coefficients  of  x  and  y  in  the  same  order  as  they  appear  in 
the  given  equations,  while  in  each  case  the  numerator  is  obtained 
from  the  denominator  by  writing  the  right  hand  members  in  place 
of  the  coefficients  of  the  quantity  ivhose  value  is  sought.  The  results 
in  this  form  are  easv  to  remember  and  convenient  to  use. 


DETKRMINANTS.  45 

Ex  \MPLES. 

\  3.V—  y=    7 


i  2.r+3j'=i2 


Here  .v= 


7 

-I  1 

12 

3  1 

3 

-I 

2 

3 

(—12)        33 


2 

3l 

\3 

7 

1  2 

12 

3 

-I 

2 

3 

9— (—  2)      II 

36 14   _     22 

9— (—2.)  ~     I  I    = 


In  this  manner  find  the  values  of  .v  and  i'  in 

<^  3-t'+7J  —  22  (3A'+27=i6 

■  (  2.i-+5j'=i5  ^-  (  7--^'+5i'=38 

I  3.V— 7)'=io  ^    i  3-^'+5,i'=i3 

(  5-^'— 3J=   2  ^'  (  7-^'+3J'=i3 

2.  Definitions.      In  the  determinant 

the  four  quantities  <;?,,  (6|,  a,,  d.,,  are  called  Elements.'^- 

The  elements  composing  any  horizoyital  line  are  called  a  Row, 
and  those  composing  any  vertical  line  are  called  a  Column. 

When  the  determinant  of  two  rows  and  two  columns  is  ex- 
panded, i.  e.  written  out  in  the  form  «,A, — a.fi^  each  term  is 
seen  to  contain  two  factors  and  hence  the  determinant  is  said  to  be 
of  the  Second  Order.  When  expanded,  each  term  is  seen  to  con- 
tain one  and  only  one  element  from  each  row  and  one  and  only  one 
from  each  column . 

3.  Let  us  now  find  the  values  of  .v,  y  and  z  from  three  equa- 
tions of  the  first   degree    containing    three   unknown    quantities. 

^7,.r-f/vr+r, -=«',.  (i). 

a,x-\-b.,y^c.,z=^d.,.  (2). 

a..x-\-b..y-\-c..z^d...  (3). 

*  These  quantities  are  called  Constituents  in  Salmon's  Modern  Higher  Algebra,  in 
Chrystal's  Algebra,  in  Aldis'  Algebra,  in  Bnrnside  and  Pauton's  Theory  of  Equations,  and 
in  'rodhiinter's  Tli-ory  of  Equations, 


46  ADVANCED     ALGEBRA. 

Multiply  (I)  by    b^c.^ — bj:.,,     (2)    by  — {b^c.^ — b.f^)    and  (3)    by 
b/,- — A/"j  and  we  obtain 

a ,  (  b.,c.^ — b/.^x + b^  (  b^c.,^ — b.f.^y^r  c^  (  b^c^ — b^c^)z=  d^  (  b^c^ — b.^c^ )        (4) . 
—a.Xb^c.—b.f^)x—blb^c—b.f;)y—clb^c—b^c^z=—dlb^c.—^^        (5). 
«,(/^jr,— V,)-^i"+Ai('^i^,— Vi)J'+^/'^i^2— Vi)-=«^^C'^i^2— Vi)        (6). 
Adding  (4),  (5)  and  (6)  we  obtain 
[«,  ( b.f.—b.x.,  )—a,{b^c.—b.f^)  +  a.Jyb^c.^—b.f^)\x 

=d^{b.f—bf.^—d^{b^c,—b..c^  )  +  d.Ab^c.—b.,c^) 

d^{b./..—b./.,)—d.,(b/.^—b.f^)-\-d.ib^c.^—b.f^) 


(7)- 
(8). 
and 


a^{  A/... — b./., ) — a,{b^c.^ — b./^  )-\-<^J  b^c., — A/, ) 
Similarly  multiply  (i)  by  — (a.f.^ — ^/a^'    (2)  by  a^c- 
(^3)  ^3'  — (^^1^2 — '^■f\)^  si^d  add  the  resulting  equations  and  we  obtain 
— d^{a.f.^ — a..c.^)-\-d.^{a^c.^ — a/J — d^{a^c^ — a/,) 


— b^  {a.f^ — a.,c.,)  +  b^{a^c^ — a/,) — b.Jya^c.^ — a/,) 
Again,  multiply  (i)   by  aj).^ — a.Jb.,,  (2)  by  - — {a^b.— 


(9). 


./, ),     and 

(3)  by  «/., — «./i,     and  add   the   resulting   equations  together  and 
we  obtain 

d^{ajK^ — a.^b.^) — d.^iaj).^ — a^b^) — d^{a^b.^ — «./,) 

z= — ' '■ — (10). 

^li^iK — ^i^^ — (^2(<^\K — "■ib^}  +  cj,a^b^ — a.,b^) 
If  we  remove  the  parentheses  from  the  denominators  in  equa- 
tions 8,  9,  10,  each  of  these  denominators  is  easily  seen  to  be 

Now  this  expression  (11)  which  contains  only  the  coefficients  of  a, 
I',  z  in  equations  (i)  (2)  (3),  we  wish  to  write  in  the  form  of  a 
square  array  where  the  letters  appear  in  the  same  order  as  in  the 
three  given  equations,  if  it  is  possible  to  interpret  such  a  square  ar- 
ray so  as  to  give  the  same  result  as  the  one  here  written.  We 
must  then,  if  possible,  interpret 
a, 

SO  as  to  give  the  same  result  as  (11).     The  quantities  composing 
the  square  array  will  be  called  Elements,  any  horizontal  line  will  be 


a^ 

K 

c^ 

«2 

K 

^2 

«3 

K 

^3 

(12) 


DETERMINANTS.  47 

called  a  Row  and  any  vertical  line  a  Column,  and  the  square  array 
itself,  or  its  equal  (11),  will  be  called  a  Determinant.  As  each 
term  of  (11)  is  the  product  of  three  quantities,  the  determinant  is 
said  to  be  of  the  Third  Order.  Now  it  is  easy  to  see  that  the  ex- 
pression (11)  can  be  obtained  from  (12)  by  taking  the  algebraic 
sum  of  all  the  products  obtained  by  taking  one,  and  only  one, 
element  from  each  row,  and  one,  and  only  one,  from  each  column; 
using  the  sign  -f  or  —  before  each  product,  according  as  the  order 
of  subscripts  in  that  product  is  obtained  from  the  natural  order 
I,  2,  3  b\'an  even  or  an  odd  number  of  interchanges  of  consecutive 
subscripts.      Hence  we  may  write 

«.,  h.,  c,    =a^b.,c.. — a^b^c.,-\-a.,b.f^ — a.,b^c..-\-a^b^c., — aj).,c^.       (13) 
I  «.  b.^  ^,. 
Each  of  the  denominators  in  (8),  (9)  and  (10)  is  equal  to  the 
determinant  ( 12);  and,  since  in  (8)  the  numerator  is  seen  to  differ 
from  the  denominator  only  in  having  «j,  a.„  a^  replaced  by  d^,  d^, 
d..,  therefore  the  numerator  in  (8)  is  equal  to  the  determinant 

d,  b,  c, 
d,  b^  c^ 
d,  b,  c. 

As  the  numerator  in  (9)  is  seen  to  differ  from  the  denominator 
only  in  having  b^,  b.,,  b.„  replaced  by  a',,  d„  d.^,  therefore  this 
numerator  equals  the  deter minint 

I  «,  d^  <:,  i 

I  'h  d,  <-. ' 

I  a.,  d.^  c^  \ 

And  similarly  the  numerator  in  the  value  of  z  in  equation  (10) 
equals  the  determinant 

a,   b,  d^ 

a.-,   b.,  d.^ 

a„  b..  d„ 


48 

ADVANCED     ALGEBRA. 

Hence  we  have 

d\   b^  cA                «!  ^1  c^ 

^1  -^i  d,\ 

d.,  b.,  c,  j               1  a.,  d^  c^ 

a,  b,  d. 

d,  b.,c.^              1  a,  ^3  c^ 

«3  K  d. 

-^  = 

^:   K  c^   ^~ 

aj  b^  Tj 

a^  b,  c. 

^■l     K     ^2 

fl,  /^,  r. 

a.,_  b,  c. 

«.     ^3     ^. 

a.,  <^3  ^., 

a.^  b.^  c.^ 

It  is  to  be  notic 

^ed  that  in  these  values  of  x,  jk  and  z  the  denom 

inator  in  each  case  is  the  determinant  formed  by  taking  the  coeffi- 
cients of  X,  y  and  z  in  just  the  order  in  which  they  appear  in  the 
three  given  equations,  and  that  in  each  case  the  numerator  is  ob- 
tained from  the  denominator  by  substituting  therein  the  right-hand 
members  of  the  equations  in  place  of  the  coefficients  of  the  quan- 
tity whose  value  is  sought. 

Examples.      Find  by  determinants  the  values  of  .v,  y  and  z 
in  the  following  equations: 

( +1-+   6r— 3.-=i7.  1    x+  j'-h2.-=34. 


-r+ 


35- 


3- 


(  5a--f-i3)'-f  4.2-=82. 
(  .r+  2>/-|-35'=  ( 
-  .r+  3.r+4--=  I 
(2.1-+    5,'+8,c-=p 


2.  -     .r+2r-f   5=33. 
(2.V+  y+    -=32. 

\    -^'+3J'+   5~=4- 
^.  -  2.i-+5r+   3-"=- 2. 
(  3A--f  9_rH-io~= 

5,  There  is  no  difficulty  in  extending  the  process  above  em- 
ployed to  a  greater  number  of  equations,  but,  after  these  illustra- 
tions, we  prefer  to  treat  determinants  by  themselves,  independent 
of  their  application  to  a  set  of  linear  equations. "^  We  will,  how- 
ever, return  to  the  solution  of  a  set  of  linear  equations  when  we 
have  obtained  a  number  of  properties  of  determinants. 

We  have  seen  how  a  determinant  of  the  second  order,  com- 
posed of  four  elements,  is  written: 

I  «,    b^  I 

I  ''.    ^  I 
and  defined  as  being  rt, A, — a.b^;    also   how    a    determinant    of  the 
third  order,  composed  of  nine  clenienls,  is  written: 


Linear  equations"  are  eqiiati.ms  of  first  degr 


DETERMINANTS.  49 

a.,  b.^  c 
a.^  b  .^  c 
and  defined  as  being  the  expression, 

Similarly  a  determinant  of  the  7zth  order  is  composed  of  n^  quanti 
ties,  called  elements  and  naturally  written: 
!  a,  b,  c,  d^  .    .  /, 

a.,  b,  c,  d.,  .    .  /, 

a.,  h..  c.  d.^  .    .  l.^ 

a,  b,  c,  d,  .    .  /, 


a^^  b^^  r,  d^^  .    .  /_ 

The  elements  in  a  horizontal  line  are  called  a  row  and  those 
in  a  vertical  line,  a  column. 

This  determinant  is  defined  as  being  equal  to  the  algebraic 
sum  of  all  products  that  can  be  formed  by  taking  one  and  only 
one  element  from  each  row  and  one  and  only  one  from  each 
column,  the  sign  +  or  —  being  written  before  each  product  or 
term  according  as  the  order  of  subscripts  in  that  term  is  derived 
from  the  natural  order  by  an  even  or  an  odd  number  of  inter- 
changes of  successive  subscripts,  it  being  understood  that  the  let- 
ters preserve  the  natural  order. 

The  collection  of  terms  written  out  according  to  this  defini- 
tion is  called  the  Expansion  of  the  determinant. 

6.  It  is  to  be  noticed  that  the  elements  are  here  represented 
b}^  letters  with  various  subscripts.  Obviously  other  symbols 
might  have  been  chosen  to  represent  the  elements,  but  the  ad  van 
tage  of  this  notation  c(msists  in  the  fact  that  the  position  of  each 
element  is  indicated.  The  letter  shows  the  column  and  the  subscript 
the  row  to  which  any  given  element  belongs.  Take  for  example 
the  element,  <f,;  since  </ is  the  fourth  letter,  the  element  belongs 
in  the  fourth  column  and,  the  subscript  being  3.  it  belongs  in  the 
third  row;  thus  its  position  in  the  determinant  is  completely  in- 
dicated.    Of  course  the  position  of  an  element  would  not  be    thus 


50  ADVANCED     ALGKBRA. 

indicated  if  there  should  be  any  disarrangment  in  the  rows  or 
columns  of  the  above  determinant,  unless  we  knew  exactly  what 
disarrangement  had  taken  place,  so  that  we  could  allow  for  it. 

7.  Since  from  the  above  definition  each  term  must  contain 
one  and  only  one  element  from  the  first  column,  therefore  each 
term  must  contain  a  with  some  subscript;  and  since  each  term 
must  contain  one  and  only  one  element  from  the  second  column, 
therefore  each  term  must  contain  b  with  some  subscript.  In  the 
same  wa}'  it  follows  that  each  term  must  contain  c  with  some  sub- 
script and  so  for  the  other  letters  Hence  all  the  letters  appear  in 
each  term  of  the  expansion  and  no  letter  appears  more  than  once 
in   the    same  term. 

Again,  from  the  definition,  each  term  in  the  expansion  must 
contain  one  and  only  one  element  from  the  first  row;  therefore 
each  term  must  contain  some  letter  with  a  sub.script  i ;  and  since 
each  term  contains  one  and  only  one  element  from  the  second 
row,  therefore  each  term  must  contain  some  letter  with  a  sub- 
script 2;  and  in  the  same  way  each  term  must  contain  some  letter 
v/ith  the  subscript  3,  and  so  for  the  other  subscripts.  Hence  all 
the  subscripts  appear  in  each  term  of  the  expansion  and  no  sub- 
script appears  more  than  once  in    the  same  term. 

From  this  it  is  seen  that  every  term  in  the  expansion  of  the 
determinant  contains  all  the  letters  a,  b,  r,  .  .  /with  all  the  sub 
scripts  I,  2,  3  .  .  «,  and  if  we  please  we  may  keep  the  letters  in 
their  natural  order  and  the  subscripts  will  be  attached  to  the.-e 
letters  in  every  possible  order.  To  illustrate,  a  dtteiminant  of 
the  fourth  order 

.?,    ^,   r,   d^ 

\a.^  Ik  c  < 
a..  A,  ('..  d.^ 

a,   b^  (\  d^ 
expanded  becomes 

a^b.f/l^—a ,  bj\d..^—a ,  b./.,d  ^  +  a ,  bj\d..  -f  <? ,  b/:,d,—a ,  b^cd, 
— a.,b^(:.d^  +  a.,b^c\d,  +  aju^d^ — ^"Afi'A — <'■,'''/',''':•,+  ''■/'4'VA 
-f  a.^b^c.d, — (i::b,i\d., — a..b/^d^  +  (^A/\^\  +  ^^J'A^'  ~  ''^A/Ai 
— a  ,b,c,d..  -\-  a ,  b,c,d,  -\-  a ,  b.,i\  d  —  a ,  />,/■,//,  — a ,  />./-,  d.,  -\-  a ,  bj  .,d^ 


5^^- 


iry^ 


I 

9 

3 

4 

I 

2 

4 

3 

I 

4 

2 

3 

4 

I 

2 

3 

4 

2 

I 

3 

4 

2 

3 

I 

db:terminants.  51 

In  this  expansion  the  signs  are  determined  according  to  the 
definition  by  the  order  of  subscripts.  Take  for  example  the  term 
afi.,c,d^,  where  the  subscripts  appear  in  the  order  4231.  This 
order  can  be  determined  from  the  natural  order  as  follows: 

Natural  order 
First,  interchange  4  and  3 
Second,  interchange  4  and  2 
Third,  interchange  4  and  i 
Fourth,  interchange  2  and  i 
Fifth,  inte: change  3  and  i 

As  5  is  an  odd  number,  the  sign  before  aj?.f.^d^  must  be  — . 

8.  The  rule  for  determining  the  sign  of  an}'  term  in  the  ex- 
pansion of  a  determinant  ma}'  be  simplified  by  noting  that  the 
interchange  of  any  two  numbers,  however  far  removed,  is  the 
sime  as  an  odd  number  of  interchanges  cf  successive  numbers. 
For  suppose  any  number  of  numbers, 

1234....//....;;/.... 

Let  there  be  /'  numbers  between  //  and  in.  Then  if  we  wish  to 
interchange  m  and  //  we  have  to  pass  m  to  the  left  successively 
over  the  r  intervening  numbers,  and  then  over  the  li:  we  have 
next  to  pass  li  to  the  right  over  the  r  numbers  that  originall}'  sep- 
arated// and  w.  In  passing  in  to  the  left  we  have  made  r-f  i 
interchanges  of  successive  numbers,  and  in  passing  h  to  the  right 
we  have  made  r  interchanges,  so  in  all  there  are  2r-\- 1  inter- 
charges  of  successive  numbers,  and  this  is  an  odd  number  what- 
ever be  the  value  of  r.  We  may  then  strike  out  the  yNOx^siiccessive 
in  the  rule  for  determining  the  sign  of  any  term  in  the  expansion 
of  a  determinant. 

The    order  4231,    derived   by  five  succesive  interchanges, 
may  be  derived  by  a  single  interchange  of  4  and  i. 

9.  By  using  subscripted  letters  for  the  elements  of  a  deter- 
minant it  can  be  expanded  with  considerable  ea.se,  but  how  can 
the  terms  be  written  out  when  other  symbols  are  used  to  denote 
the  elements?     vSuppose  we  wish  the  expansion  of 


52  ADVANCED     ALGEBRA. 

a     b     c 

d    e     f  (i). 

g     h     k 
Now  we  do  know  the  expansion  of 

a,     K     '-'■z  (2). 

a.,     d^     q  ! 

It  is  a^djC.^ — a^d^c, — a.,b^c.^-\-a.,b^c^  +  a^l>^c^_ — a^b./^,  and  to  obtain 
the  expansion  of  (i)  we  must  of  course  substitute  for  each  of  the 
elements  in  (2)  that  one  which  in  (i)  occupies  the  same  position. 
The  expansion  of  (i)  then  becomes 

aek — a/if- — dbk-\-dhc-\-gbf^get. 
A  better  method  will  be  given  further  on. 

10.  Theorem.  The  expajisioji  of  a  determinant  of  the  n\\\. 
order  contains  123.    .n  terms. 

From  the  definition  the  terms  are  obtained  by  taking  the  let- 
ters a,  b,  c,  .  .  .  in  their  natural  order  and  the  subscripts  in  every 
possible  order.  Whence  the  number  of  terms  is  the  same  as  the 
number  of  ways  of  arranging  n  things  taking  all  at  a  time,  which 
is  123  .    .  n. 

11.  Theoreij.  If  in  any  determinant  the  rows  are  changed 
into  colum7is  and  vice-versa ,  the  determinant  is  not  changed. 

Thus  we  have 


I  fl, 

/;, 

(^ 

d. 

1  '^■l 

b, 

c, 

d, 

•  ^., 

b. 

c, 

d. 

!  ^4 

b, 

c. 

d 

b. 


,     ,       .       ,       ,  I  \  d,    d]     d.^      d^  i 

For  convenience  represent  the  first  determinant  by  J  and 
the  second  by  J'.  Now  it  is  evident  that  J' involves  the  subscripts 
in  the  same  way  that  J  involves  the  letters  and  vice  versa.  J' 
equals  the  sum  of  all  the  products  formed  by  taking  one  and  only 
one  element  from  each  row  and  from  each  column,  hence  the 
terms  in  the  expatision  of  J'  are  the  same  as  those  o^  J.  More- 
over the  signs  are  alike  because  the  signs  prefixed  to  the  te-ms  of 
-I'  are  determined  by  the  number  of  interchanges  of  letters  from 
natural  order  abed  while  the  signs  prefixed  to  terms  of  J  are  deter- 
mined by  the  number  of  interchanges  of  sub.scripts  from   the  nat- 


DETERMINANTS.  53 

ural  order  1234,  and  it  requires  the  same  number  of  interchanges 
of  letters  to  pass  to  a  given  order  as  it  does  of  subscripts  to  pass  to 
a  corresponding  order. 

Thus  a.fi^c^d.^  or,  what  is  the  same  thing,  d^a,d./^,  is  a  term  of 
both  determinants  and  it  evidently  requires  the  same  number  of 
interchanges  of  letters  to  pass  from  order  adcd  to  the  order  bade  as 
it  does  of  subscripts  to  pass  from  order  1234  to  order  2143. 

12.  From  this  it  follows  that  any  theorem  that  can  be 
stated  about  the  rows  of  a  determinant  can  be  stated  about  the 
columns,  and  z'ice  versa.-  for  we  may  write  side  by  side  two  deter- 
minants in  which  the  rows  of  one  are  the  columns  of  the  other 
and  any  theorem  about  the  rows  of  one  is  a  theorem  about  the 
columns  of  the  other. 

13.  Theorem  If  hvo  nnvs  or  two  columns  of  a  determinant 
arc  interchanged  the  sign  of  tJic  determinant  is  changed. 

Consider  the  two  determinants 

I     «,  /7,         ,,        d^ 

I  a.,     d.,     c,    d., 

I  a'.     /;■     c.     f 

I  ^^     ^     <i    '^'. 

Represent  the  first  by  J  and  the  second  by  J',  J'  being  de- 
rived from  J  by  interchanging  the  second  and  fourth  rows.  Now 
take  any  term  of  J'  as  a^  l\  c,  d..  This  is  found  in  J  and  as  in  the 
expansion  of  -1  the  term  a.,  b^  <■;  d.^  has  the  sign  +.  So  in  the  ex- 
pansion of  J'  the  term  a^  h^  c,  d..  has  the  sign  -|-  but  a^  b^  c^  d^  is 
also  a  terra  of  J  and  as  a  term  of  J  it  has  the  sign  — .  In  the 
same  way  every  term  in  the  expansion  of  J'  appears  in  the  expan- 
sion of  J  with  an  opposite  sign;  therefore  J= — J'. 

General  Case.  Represent  the  determinant  of  the  wth  order 
by  J  and  from  J  form  another  determinant  by  interchanging  the 
/v'th  and  rth  rows  and  represent  the  new  determinant  by  J'.  Now 
select  a7iy  term  in  the  expansion  of  J'  differing  from  the  selected 
term  of  J  only  in  having  the  -^th  and  rth  subscripts  interchanged, 
the  sign  being  the  same  in  each  case.  This  term  in  the  expan- 
sion of  J' also  appears  in  the  expansion  of -I  but  with  an  opposite 


and 


b,       c 

\     ^U 

b^      c 

\     d 

b..     c 

:.     d 

b.,     i 

•.,     d 

54  ADVANCED     ALGEBRA. 

sign,  being  determined  from  the  previously  selected  term  of  the 
expansion  of  J  by  interchanging  two  subscripts,  which  of  course 
changes  the  sign.  In  the  same  way  every  term  in  the  expansion 
of  J'  is  found  in  the  expansion  ot  J  with  an  opposite  sign.  There- 
fore -l=— J'. 

14.  Corollary.  If  a  determinant  has  tivo  rows  or  two  col- 
umns  identical  the  determinant  equals  zero.  For  if  we  interchange 
the  two  identical  rows  or  columns  in  the  determinant  represented 
by  J,  we  get  a  determinant  represented  by  — J  ;  but  interchang- 
ing two  identical  rows  or  columns  cannot  change  the  determinant 
at  all.      Therefore  J=— J  ;  2J=o  and  J=o. 

15.  MikoR  Determinants.  "  If  in  any  determmant  we 
erase  any  number  of  rows  and  the  same  number  of  columns,  the 
determinant  formed  with  the  remaining  rows  and  columns  i^ 
called  a  Minor  of  the  given  determinant.  The  minors  formed  by 
erasing  one  row  and  one  column  are  called  first  minors;  those 
formed  by  erasing  two  rows  and  two  columns  are  called  second 
minors,  and  so  on." — Salmoyi  s  Modern  Higher  Algebra. 

If  the  given  determinant  be  of  the  «th  order,  the  first  minors  are 
of  the  {n — i)th  order,  the  second  minors  are  of  the  [n — 2  )th  order, 
and  so  on:  so  we  may  speak  of  rth  minors  or  minors  of  the  order 
n — r  indifferently.  Minors  of  the  first  order  are  the  elements 
themselves. 

The  elementsat  the  intersection  of  the  rows  and  columns  erased 
also  form  a  minor  of  the  given  determinant  called  the  Complemen- 
tary of  the  minor  which  is  left.  In  any  determinant  the  comple- 
mentary of  an  rth  minor  is  a  minor  of  the  rtli  order.  If  the  de- 
terminant is  of  the  nth  order  the  complementary  of  a  minor  of 
the  order  r  is  a  minor  of  the  order  ;/  —  r,  or  the  complementary  of 
an  rth  minor  is  an  {n—r)  th  minor. 

In  any  determinant  there  are  as  many  ist  minors  as  there  are 
elements  and  the  first  minors  obtained  by  erasing  the  row  and 
column  intersecting  in  any  given  element  is  called  the  first  minor 
corresponding  to  that  element.      In  the  determinant 


DETERMINANTS.    •  55 

«:,        A3       ^3 

which  we  represent  by  J,  if  we  erase  the  first  row  and  first  column 
there  remains  the  determinant  I  b.^  c^  I  which  is  the  first  minor  of 

J  corresponding  to  «,,  and  similarly    I  Z",  <:,  I     is  the  first  minor  of 

I  ^  c,  I 
J  corresponding  to  a.,;  also    I  a,  q  I    is  the  first  minor  of  J  corres- 

I   ^,  ^:i    I 

ponding  to  b^;  and  soon. 

16.  Expression  of  a  Determinant  in  Terms  of  the 
Elements  of    any    Row   or  Column. 

j^,  b^  r,  d^ 

I    ^2.    ^2     ^2     ^2 

I  a.^  b^  <:,,  d^ 

\<^^    K    ^,    A 

Represent  the  above  determinant  by  J.  Since,  by  the  defin- 
ition of  a  determinate,  every  term  in  the  expansion  must  contain 
some  element  from  the  first  column,  a  certain  number  of  these 
terms  will  contain  a,,  while  other  terms  will  contain  a.^  and  so  on. 
Collect  together  all  the  terms  of  the  expansion  of  J  which  contain 
(7,,  and  after  taking  out  this  common  factor  «,,  there  will  remain 
an  aggregate  of  terms  which  we  will  represent  by  A^,  so  that  a^A^ 
will  represent  the  algebraic  sum  of  all  those  terms  which  contain 
a,.      Referring  to  Art.  7,  we  see  that 

A=b/.^d^ — b.f^d.^ — b^c.^d^-\-  b.f^d.,-\-b^c.^d.^ — ^3^2- 

In  the  same  manner  we  might  collect  together  those  terms  in 
the  expansion  of  J  which  contain  the  element  a.^  and  the  algebraic 
sum  of  all  these  terms  would  be  represented  by  a^A^  and  from 
Art.  7  A.,=  — b^cj^+b^c^d^+b.^c^d^ — k.f^d^—bf^d.^-\-b^c^d^. 
Similarly  the  algebraic  sum  of  all  the  terms  in  the  expansion  of 
J  which  contain  a^  would  be  a.^A.^  and  the  algebraic  sum  of  those 
terms  containing  a^  would  be    a^A^. 

Now  as  every  term  in  the  expansion  of  J  must  contain  some 
element  from  the  first  column  and  if  we   collect  into   one    group 


56  ADVANCED     ALGEBRA. 

those  which  contain  a^  and  into  another  group  those  which  con- 
tain a.^  and  into  another  those  which  contain  a,  and  into  another 
those  which  contain  a^  ;  then  in  these  four  groups  we  will  be  sure 
to  have  all  the  terms  of  J. 

Therefore,  A=a^A^-Ya.,A.,-^a..A..^^a^A^. 

This  is  an  expression  for  J  in  terras  of  elements  of  the  first 
column;  in  a  similar  way  we  could  have  obtained  an  expression 
for  J  in  terms  of  the  elements  of  the  second  column  or  any  other 
column  or  any  row. 

If  we  select  the  terms  in  the  expansion  of  J  which  contain 
any  one  of  its  sixteen  elements  and,  after  taking  out  this  common 
factor,  represent  the  remaining  aggregate  of  terms  by  a  capital  let- 
ter of  the  same  name  and  with  the  same  subscript  as  the  element 
we  are  considering:  then  J  may  be  expressed  in  any  one  of  the 
following  eight  ways: 

A=a^A^+a.^A.,-\-a.^A,^-\-a^A^.  (i). 

J=  b^B,  +  bfi.^  +  b,B,,  +  b,B,,  (2 ). 

J=  ,-,  C,  +  cC  +  qC,  +  r^C,.  (3). 

J  =  d^l)^  -j-dM.^-hdM,-j-d^D^  (4). 

J=rt,.-i, +/7,^",+  f,r, +  </>,.  (s). 

J=a,A.,  -f  b.,B.+  c,  C+d^n.,.  (6). 

J=a^,Al  +  b,B:  +  'c,,C,-^d,D,.  (7). 

A  =  a^A^  +  bJ^^  +  r,C,  +  <A-  C-*^)- 

The  explanation  is  here  given  for  a  determinant  of  the  fourth 
order.  It  is  so  evident  that  the  process  applies  to  a  determinant 
of  any  order  that  we  omit  a  separate  explanation  for  the  general 
case. 

17.  Expression  of  a  DE;TERMrN.\.xT  ix  Terms  of  the 
First  Minors  Corresponding  to  any  Row  or  Column. 

As  we  usually  represent  a  determinant  by  J,  so  let  u-;  repre 
sent  the  first  minor  corresponding  to  «,  by  -J«i  and  similarly  repre- 
.sentthe  first  minor  corresponding  to  any  element  by  J  with  that 
element  used  as  a  subscript. 

We  will  prove  that  in   the  eight    equations  above    the   factor 


DETERMINANTvS.  57 

that  multiplies  any  element  is  either  +  or  —  the    first  minor  cor- 
responding to  that  element. 

First,   to  prove      .-J,=  -''',. 

The  terras  of  .4 J  are  obtained  from  the  terms  of  -I  that  con- 
tain the  element  a,  by  striking  out  this  element;  that  is,  they 
consist  of  the  letters  ^,  r,  ^/  in  the  natural  order  with  the  sub- 
scripts 2,  3,  4  attached  to  the  letters  in  every  possible  order. 
Moreover  in  those  terms  of  J  from  which  the  terms  of  A^  are  de- 
rived, the  element  a,  stands  at  the  head,  and  hence  the  sign  is  de- 
termined by  the  luimber  of  interchanges  of  the  /ast  three  sub- 
stripts.  But  the  terms  of  -l«j  also  consist  of  the  letters  b,  c,  d  with 
the  subscripts  2,  3,  4  attached  to  these  in  every  possible  order  and 
the  sign  of  each  term  is  here  also  determined  by  the  number  of 
interchanges  of  the  subscripts  2,  3,  4;  hence,  ^-4,  =  J",. 

Second,   to  prove     A.,-^  —  ^a,^. 

The  terms  of  A.,  are  determined  from  the  terms  of  J  that  con- 
tain the  element  a,  by  striking  out  this  element;  that  is,  they  con" 
sist  of  the  letters /^  ^,  i/ vi^ith  the  subscripts  i,  3,  4  attached  to 
these  letters  in  every  possible  order.  The  sign  of  any  of  these 
terms  is  the  same  as  the  sign  of  the  corresponding  term  in  the  ex- 
pansion of  J  and  in  this  term  of  J  it  requires  one  interchange  of 
subscripts  to  begin  with  to  get  the  subscript  2  or  the  element  a^  at 
the  head,  and  so  the  sign  of  any  term  of  A.,_  is  determined  by  a 
number*  one  greater  than  the  number  of  subsequent  interchanges 
in  the  subscripts  i,  3,  4. 

The  terms  of  J"?,,  also  consist  of  the  letters  b,  c,  d  with  the 
subscripts  i,  3,  4  attached  to  the  letters  in  every  possible  order; 
but  the  sign  of  any  term  of  Ja.^  is  determined  by  the  number  of 
interchanges  of  subscripts  i,  3,  4  ;  hence  the  sign  of  any  term  in 
J".,  is  opposite  to  the  sign  of  a  corresponding  term  in  A^_ ;  and 
consequently  .-V.,= — -!«,, 

■■Plus  if  tliis  miinliei'  is  even  and  iiiimis  if  tliis  number  is  odd.  . 


58 


ADVANCED     ALCxEBRA. 


In  exactly  the  same  way  it  may  be  shown  that 


^3= +  4; 


A  =  — J^,   - 


B,=  +  J/u ,  etc. 


Hence  we  see  that  the  nmltipUer  of  any  element  in  the  ex- 
pansion of  a  determinant  is  either  +  or  —  the  first  minor  corres- 
ponding to  that  element,  the  sign  +  or  —  being  used  according  as 
the  element  is  removed  an  even  or  odd  number  of  steps  from  the 
element  in  the  upper  left  hand  corner  ;  where,  in  counting  t'-.e 
steps,  we  pass  along  the  first  row  to  the  right  until  we  are  in  the 
column  in  which  the  element  is  found,  and  then  downward  until 
we  come  to  the  element,  but  never  pass  along  a  diagonal  line. 

Calling  that  diagonal  running  from  the  upper  left  hand  corner 
to  the  lower  right-hand  corner  the  I^eading  or  Principal  Diag- 
onal, then  the  rule  just  given  to  determine  the  sign  may  be  simpli- 
fied. The  sign  +  or  —  is  used  according  as  the  element  taken 
is  an  even  or  an  odd  number  of  steps  from  any  element  in  the 
principal  diagonal. 

All  this  is  given  for  a  determinant  of  the  fourth  order,  but  a 
careful  examination  of  the  discvission  will  show  that  it  is  equally 
applicable  to  a  determinant  of  the  ;^th  order. 

1 8.  The  above  gives  us  a  new  way  of  expanding  a  deter- 
minant. Take  for  instance  the  determinant  of  fourth  order  and 
express  it  in  terms  of  first  minors  corresponding  to  the  elements 
of  any  row  or  column;  say  the  first  column: 


«,  ^:  ^,  ^1 

b,  ^■,  d., 

J= 

a.,d.,c.,d^  =ci, 

b,c,d. 

iicM 

b,  c,  < 

a^  b^  <r,  d^ 

b.,  r..  < 

b,  c,  d, 

b,c,d. 

^a,   b.,c^d., 

b,c,d, 

■    bj,d, 

b,  c,  d, 
b.,c,d, 
bled.. 


Now  we   can  expand  each  of  these  determinants  of  third  order 
in  terms  of  their  first  minors  in  the  same  way: 


DETERMINANTS. 


59 


J=^ 


-a,  I  b, 
+  a.,  {  b. 


•«.  I  ^ 


c^  <  I 
c,  d.,  I 

c.dA 
c,  <  I 
c.dA 
c.  d.. 


d., 
d] 
d, 

<  I  I  ^:, 

<  I  +'^.  I  '-. 

<  I        k. 
^1 1  +^'  I  ^, 


d,  I 

d,  I 

^,  I 
^,  I 
^,  I 


Each  of  these  twelve  determinants  of  the  second  order  we  can 
expand  and  we  then  have  the  expansion  of  the  determinant  -J  as 
follows: 

I       a  J  b^c^d^ —a^b.,c^d.^ — a ,  b.f.^d^ + a^b^c^d.^  -\-  a ,  b^c^d.^ — a ,  b^c^d^ 

I _  J  — ^iPx^id^ + aj)^c^d^ + a.p.f^d^ — aj}.f^d^ — a.p^c^d.^-\-a.p^c.,d^ 

~  I  +  afi^c^d^ — a.Jb^c^d^ — a.Jt\c^d^ + ajy,^c^d^  -\-  a.b^c^d^ — a.pf.^d^ 

I  — aj)^c.4.^  +  ajj^c^d^  ■\-  aj).f^  d^ — ajb^c^d^ — ajb^c^d., + ap.^c^d^ 

The  result  agrees  with  the  expansion  given  in  Art.   7. 

By  the  process  here  given  we  can  expand  a  determinant  when 
the  elements  are  represented  by  any  symbols  whatever  as  easily  as 
when  the  elements  are  represented  by  letters  with  subscripts:   thus 

a 
d 
g 

=a(ek — hf)—d{bk—hc)  +g-{b/~ec) 
=  aek — ahf^dbk-\-dhc-\-gbf—gec. 
19.   Examples.     /.     Express  the  determinant 

a     b      c 

d     e     f 

,g     h     k 

in  terms  of  the  first  minors  corresponding  to  the  second  column 
and  expand  the  resulting  determinants  of  second  order  and  show 
that  the  result  agrees  with  that  given  in  the  last  article. 

2.  Express  the  same  determinant,  in  terms  of  minors  corres- 
ponding to  the  elements  in  the  second  row  and  show  that  the 
final  result  is  the  same  as  before. 


c 

f 

=  a 

e     f 
h     k 

—  d 

b     c 
h     k 

+  g 

b     c 
c    f 

k 

6o  ADVANCED     ALGEBRA. 

J.     Find  the  value  of  the  following  determinants 


2 

3 

I 

3 

I 

I 

2 

3 

I 

3 

2 

3 

3 

2 

3 

4 

I 

4 

4 

I 

4 

4 

4 

4 

3 

3 

4 

3 

3 

3 

3 

3 

3 

20.  Theorem.  If  all  the  elements  of  any  roiv  or  column  can 
be  expressed  as  the  sum  of  two  or  more  quantities,  then  the  deter- 
minayit  can  be  expressed  as  the  sum  of    two  or  more   determinants. 

Take  for  example  the  determinant, 

I  C«i  +  '^^)   b,  c, 
(«,+/5j   b..  c, 

I  r^+;j;)   b,  q 

Expressing  this  in  terms  of  minors  corresponding  to  the  elements 
of  the  first  column,  we  get 

I  b.^  c,  I  I  b,_  c.^\         ^  I  b,,  c,  I 

^  ;   "■\  \  ^1  ^1  I  ~'^-!.  I  '^i  '"i  I  +"•:!  I  <^i  '"i  I    * 
b..  c.  b..  c,  />.,  c, 


( 


b.  C| 
b.  c„  I 


\ 

b,  c^  I   +;-..  I  b,  r,  I    ) 
b-,  c,  I  I  ^.,  C  I    ^' 


But  the  quantity  in  the  first  bracket  is  evidently  equal  to  the 
determinant 

b. 
,  b., 

.  b, 

and  the  expression  in  the  second  bracket  equals  the  determinant 


Hence 


I  (^  +  '^',)  b,  c, 
I  («..+/5,)  b,,  c. 


K   b. 

c 

/A,  b.,  c, 

K  K  ^i 

'^   b^  ^> 

>\  ^  c. 

= 

"■;       b^      C, 

+ 

K  b,  C, 

a 

A,    r. 

K  b,  c. 

DETERMINANTS. 
If  the  given  determinant  had  been 

('^.+,^.,4-;- J  A,  c. 
then,  by  what  has  just  been  given,  this  determinant 


(«,+'5,)  b^  r, 

/-,   /',  ^-, 

(S+/^J  b.  c, 

+ 

/'.  A,  .; 

(«.+/5^)  ^  ^. 

;-.,  b.  c. 

and  suppl3'ing  the  value  of  the  first  of  these  from  above   we    have 


("■,  +  /5,  +  r:)    b^ 

i";+f^A-r^  b, 
("■.+>',+rd  b. 


c^ 

a,    b^    i\ 

^2 

= 

"■■I  b.,  c, 

c, 

'h  b-f,  c.^ 

i:^ 


b, 

b, 
?,  b. 


r,  A,  ^1 

+     r,  ^,  ^-2 

r,  b.,  c. 


and  so  if  all  the  elements  of  the  first  column  were  the  sum  of  any 
number  of  quantities,  then  the  determinant  equals  the  sum  of  the 
same  number  of  determinants,  the  forms  of  which  are  evident 
from  the  example  here  given. 

Evidently  this  peculiarity  might  have  presented  itself  in  any 
other  column  as  well  as  the  first  or  in  any  row. 

A  precisely  similar  discussion  would  show  that  if  all  the  ele- 
ments of  any  row  or  column  were  expressed  as  the  difierence  be- 
tween two  quantities,  then  the  determinant  could  be  expressed  as 
the  difference  between  two  determinants. 

21.     Example.     Express  the  determinant 

2  3     I 

3  3  3 

4  4    I 

as  the  sum  of  two  determinants  in  three  different  ways;  find  the 
value  of  each  of  the  resulting  determinants  and  compare  the  sum 
with  the  value  of  the  given  determinant.  Also  express  it  as  the 
sum  of  three  determinants,  find  the  value  of  each  and  add. 


62  ADVANCED     ALGEBRA. 

22.  Theorem.  If  all  the  elements  of  any  row  or  column  be 
multiplied  by  a  common  factor  the  determinant  is  rmdtiplied  by  that 
factor. 

Take  the  following  determinant,  which  we  represent  by  J  : 

\  a  b  c 
\d  e  f 
\g  h  k 

and   multiply    all    the    elements  in    the   first  column  by   m    and 
we  obtain 

ma   b    c  I 
7nd  e  f 
?no-  h  k 

Calling  this  J',  express  J  and  J'  in    terms    of  the    minors   corres- 
ponding to  the  first  column  and  we  get 


=  <7 

c    f 
h  k 

-d      b    c 
h  k 

+^ 

b  c 
e  f 

-ma 

e  f 
h  k 

— md     b    c 
h  k 

+  mg 

b  c 
c  f 

from  which  it  is  evident  J'=wJ 

Corollary  i.  If  all  the  elements  in  any  row  or  column 
contain  a  common  factor  that  factor  may  be  taken  out  of  each 
of  the  elements  and  placed  as  a  factor  of  the  remaining 
determinant. 

Example.      Verify  the  corollary  in  this  determinant: 


I 

2 

3 

3 

3 

3 

r 

4 

4 

Corollary  2.  Multiplying  any  row  or  column  by  any 
number  and  dividing  another  row  or  column  by  the  same  number 
does  not  change  the  value  of  a  determinant. 


DETERMINANTS. 


63 


23.     Theorem.      If  the  elements  of  any  row  or  eolumn,  each 
multiplied  by  the  same  number,  be  added  to  or  subtraeted  from,  the 
corresponding     elements     of  another     rozv  or   column,    the     deter- 
minant is  not  changed.^^^ 
Take  the  determinant 

a  b  c  \ 
d  e  f  I 
g  h  k  ; 

and  add  to  the  elements  of  the  first  column  the  corresponding  ele- 
ments of  the  second  column  each  multiplied  by  m  and  we  get 

(  a-\-mb)   b    c 

I   i^d-\-me)  e  f 

Ig+mh)  h  k 

Now,  because  each  element  in  the  first  column  is  the  sum  of 
two  quantities,  therefore 


{a-\-mb)  b  c 

a    b  c 

{d-\-me)    ef 

= 

d    e  f 

-f 

(g-\-mh)  h  k 

g  h  k 

b 

b  c 

e 
h 

<^  f 
h  k 

a   b   c 

b    b   c 

= 

d  e   f 

4- 

e    e  f 

g  h  k 

h  h  k 

Taking  out  the  factor  m  from  the  elements  of  the  first  column 
of  the  second  determinant  on  the  right  side  of  the  equation  we  get 

{a-\-mb)  b  c 
(d-j-me)  e  f 
{g-\-mh)  h  'k 

But  the  last  determinant  in  the   equation    has   tv,^o   identical 
columns  and  therefore  vanishes,  whence 

I  (a-\-mb)  b  c 
I  (d+me)  e  f 
!  [g+mh]  h  k 

*rhe  wording  of  the  theorem  sliould  be  carefully  noted,   for  if  the   elements  of  any 
row  or  column  be  added  to  or  subtracted  Irom  the  corresponding  elements  of  another  row 
or   column  multiplied  by     the  same  number  the  determinant  is  changed. 
If  from  the  determinant 

'  a  h  <■   ' 

.  d  e   f 

g  h  'k 

we  make  another  by  adding  the  elements  of  the  second  column  to  111  times  the  elements  of 

the  first  column  we  get 

(;«    +b)  h  c   , 
I  mil+e  )  ,-  f 
{ mg^h  \  li  k   , 
and  this  is  just  m  times  the  first  one. 


a  b    c 

= 

d  e    f 

g  h  'k 

a    b    c 

b    b    c 

a    b   c 

=      d  e    f 

—  m 

e     e   f 

= 

d   e  f 

g  h  k 

h    h    k 

g    h  k 

ADVANCED     ALGEBRA. 
Similarly 

{a — mb)  b  c 
{d — me)  e  f 
ig—mh)  h  k 

Scholium. — When  dealing  with  a  numerical  determinant 
where  the  elements  are  large  numbers  we  may  combine  rows  with 
rows  and  columns  with  columns  according  to  this  theorem  so  as 
to  reduce  the  elements  to  smaller  numbers  and  thus  obtain  a 
determinant  easier  to  compute. 

24.  Theorem.  If  all  the  elements  but  one  in  any  row  orcol- 
um.71  be  zero  the  determinant  is  reduced  to  one  of  the  next  lower  order. 

Take,  for  example,  the   determinant 


«, 

^ 

0  d, 

a.. 

b., 

0  d. 

a.^ 

b\ 

0  < 

a. 

h^ 

e,  d. 

fl, 

b. 

d, 

a.. 

b.. 

d., 

a^ 

b^ 

< 

a 

l\ 

d^  \ 

a 

J.. 

d., 

a 

'.K 

d. 

Express  this  in  terms   of  minors    corresponding    to    elements    in 
third  column  and  it  equals 

a^  /;,  d.,  a^    b^    d^  a,    b^    d^  \  a,    /',    d^ 

a.^  b^  d.^     — o     a^    b.^    d.,      -fo     a,    b..    d^      —  r,  |  a.,    b.,    d, 
a^  b^  d^  a^    b^    d^  a.    b.    d.  I  a.,    b.,    d.. 

which  equals 

a,  b,  d 

25.  To  Compute  the  Value  of  a  Numerical  Deter- 
minant. If  we  have  to  compute  the  value  of  a  numerical  deter- 
minant it  is  well  (i)  to  see  if  all  the  elements  of  any  row  or  column 
contain  a  common  factor  so  that  as  many  common  factors  as  pos- 
sible may  be  removed  in  order  to  reduce  the  elements  to  smaller 
numbers;  then  (2)  to  seek,  by  some  combination  of  rows 
with  rows  or  columns  with  columns,  to  still  further  re- 
duce the  elements,  especially  aiming  to  transform  the  de- 
terminant     so      that      in      some    row      or     some     cohimn      all 


DETERMINANTvS. 


65 


the  elements  but   one   shall   be  zero,    when    the   determinant   is 
reduced  to  one  of  a  lower  order.     We  then   treat   the   new   deter- 
minant in  a  similar  way  and  thus  by  continual  reductions  we  may 
find  its  value  usually  much  more  easily  than  by  expanding. 
Let  us  compute  the  value  of  the  determinant 


from  last  and  we  get 


Subtract  two  times  first  column  from  second  column;  also  subtract 
fourth  column  from  third;  and  subtract  fourth  column  from  fifth: 


6    12    6    3      9 

3      631      2 

49412 

5    i^    3    2      5 

12    24    6    4   12 

Take  factor  3  from  first  row  and  2  fi 

2      4213 

3      6312 

6 

49412 

5    10   3    2    5 

6    12    3    2    6 

2  o 

3  o 

4  I 


1  I    2 

2  I    I 
1    I    I 


=   —6 


5012 
,601    2   4  I 
Subti'act  fourth  column  from  first: 

I  o   I    12] 

-6     '   '    '    '\  = 

2    I   2   3  I 

I  2    I    2  4  I 
Subtract  third  row  from  fourth : 
I  o   I    I    2 

1  I   2   I    I 
—  12 

I  I    I   2  3 

I   O    O    O    I 

Subtract  first  column  from  second; 

!  o  I 
— 12  I  I  I 


2  1 

3  2 

5  I 

6  I 


I  2 

1  I 

2  3 

2  4  I 


0  I 

1  2 
I  I 
I  I 


1 1 


I  2| 

1  I    I 

2  3l 
2  4l 

1  1 1 

2     I    I 
I     2    I 


I    I     O    2    I 


66 


ADVANCED     ALGEBRA. 


Subtract  first  row  from  second ; 


O    I 

I 

I    o 

o 

I    o 

2 

I    I 

O    2 


H 


A^ 


A..=- 


li.  c.  d. 

J,„= 

i^:  G  < 

^  ^\  < 

b^  r,   ^, 



b.,  c.  d.. 

b\  c\  d\ 

etc. 


whicli  is  the  value  of  the  determinant  of  the  fifth  order  that  we 
started  with. 

26.      In  article  16  eight   different  expressions  were  given  for 

the  determinant 

\  a^  /?,   r,  d^ 

1  a.,  b.,  c,  d., 

I  a^  b..  r.,  d^ 

each  in  terms  of  the  elements  of  some  row  or  some  column,  and  it 
was  noticed  that 

Keeping  carefully  in  mind  the  meaning  thus  given  to  the  cap- 
ital letters  with  various  subscripts  it  is  evident  that 

b^A^  +  b.^A.^+b.^^A.,_-^b^A^ 

I  b,  (\  d,  I  b^  r,  <  I  I  /;,   r,  ^,  I 

-b.,    I  b.,  q  d..  I    -\-b..     b..  c.  d.,       -bA  /'„  r,  d.,  \ 
I  b\  c^  d\  I  ■■     b\  f]  d\  !  I  b.._  a  d[  I 

and  this  is  evidenth^  the  expression  of  the  determinant 
I  /7,   b^  r,  d^ 
I  b..  b.,  c.  d.. 
I  b,  b..  <■..  d. 
I  '''.  ^',  '\  < 

in  terms  of  the  minors  corresponding  to  the  elements  of  the  first 
column.  Now  this  determinant,  having  two  identical  columns, 
equals  zero;  hence 

b^  A^  +  b.^A.^+  b.,A.^+  b^A=o. 

In  the  same  wav  we  could  obtain   a  relation  connecting   the   ele- 


b..  c,  d., 

=^ 

b.,  c.  d.. 

b.  c\  < 

DETERMINANTS. 


67 


ments  of  any  row  or  column  with  minors  corresponding  to  the  ele- 
ments of  some  other  row  or  column.  There  would  be  in  all  twen- 
ty-four such  relations  given  by  the  determinant 


rt, 

l\ 

c^ 

< 

a., 

i 

c.. 

d. 

a.. 

Ik 

c. 

cl, 

'^^ 

ih 

r. 

d, 

In  the  same  way  if  we  were  given  the  determinant  of  the  ?zth  order: 

a,  b^  c\  .    .     .    .  /,   ; 
«2  b.,  r,  .    .    .    .  /, 
a.,  b.,  c,  .    .    .    .  l. 


we  could  obtain  2;/  different  expressions  for  it,  each  one  as  multi- 
ples of  the  elements  of  a  row  or  a  column  and  we  could  obtain 
2n{,n — i)  other  relations  connecting  the  elements  of  one  row  or 
column  with  the  minors  corresponding  to  the  elements  of  another 
row  or  column.  As  samples  we  write  two  equations  of  each  kind 
and  leave  the  student  to  write  others. 


a.,A.^+b.,B.,+c.,C.,  4- 
a..A.,+  b.B.,+  cC  + 


+  a^A~A. 

a^B=o. 
l.L=o. 


Examples      /.    Write  the  six  different   expressions  for 


and  verify  each  expression. 

2  Write  the  twelve  other  equations  expressing  the  relation 
between  the  elements  of  one  row  or  column  and  the  minors  corre- 
sponding to  the  elements  of  another  row  or  colunui  of  the  determi- 
nant in  E.K.  /. 


68  ADVANCED     ALGKBRA. 

J.     Express  the  value  of 

I  a      h      r  I 

I  <^     ^      f  I 

I  K     h      k  I 
in  six  different  wa^-s. 

4.  Write  the  twelve  other  equations  expressing  the  relation  be- 
tween the  elements  of  one  row  or  column  and  minors  correspond- 
ing to  the  elements  of  atiother  row  or  column   of  the  determinant 

in  Ex.  3. 

28.  Application  TO  THE  Solution  OF  Stmut.taneous  Equa- 
tions OF  First  Degree. 

Let  us  take  ?2  equations  of  the  first  degree  containing  «  unknown 

quantities. 

a^.x-\-l\y-\-c^z-\- .    .    .    .-f/|T'=w,. 

a.^x-\-b.^y-\-c.,s+.   .    .    . -{- /-c>=  ff/ ... 

Here  we  suppose  that  the  determinant  formed  by  the  coeffi- 
cients of  the  unknown  quantities,    viz: 

a^  b,  (\  .  .  .  .  /,  i 
a.^  b,  c,  .  .  .  .  /,\ 
a.,  b.,  c,  .    .     .    .  /.,   !   is  not  zero. 

«„  "„  ^„  •    ■    •    •    ^„ 
Multiph-  the  first  equation  by  A^,  the  second  by  A.,,  and  so  on,  and 
we  have 

a.,A.^x^b..A,y+c.,A.,z^  ....  -\-/.,A.,v=7n.,A,. 
a^,A^,x+li^A^y-\-cA,z-{-  ....  -j-LA..y=nL,A,,. 

a„  +  Ax+b^A^y+c„A^2-^    .    .    +/^Av=w^A,,. 

Then  adding  we  obtain  for  the  coefficients  of  a- the  determinant 

of  the  coefficients  of  all  the  unknowns,   which    we    will   represent 

by  J;  the  sum  of  the  coefficients  of  each  of  the    other  unknowns 

become  zero  (see  Art.  25  >;  and  the  right-hand  meml)er  is  what    J 


DETERMINANTS.  69 

becomes  when  the  a's  are  replaced  by  the  n/'s,  i.  e.,  by  the  right 
hand  members  of  the  given  equations.  Let  us  represent  this 
determinant  by  J,.     Then 

•J,  =  -',;  therefore  .v= 

J, 

In  the  same  way,  if  we  multiply  the  first  equation  by  B^,  the 
second  by  B.,,  and  so  on,  and  add  the  resulting  equations,  we  get 

__  J.^ 
-'      T' 
where  J.^  means  what  J  becomes  when  the  d's  are  replaced  by  the 
right  members  of  the  given  equations. 

Again,  multiply  the  first  by  C,,  the  second  by  C,  and  so  on, 
and  add  the  resulting  equations,  and  we  get 

J 

where  J.,  is  what  J  becomes  when  the  r's  are  replaced  by  the 
right  members  of  the  given  equations. 

It  is  now  evident  that  the  value  of  an  v  unknown  quantity  in  the 
given  set  of  equations  is  the  ratio  of  two  determinants,  in  which 
the  denominator  is  the  determinant  of  the  coefficients  in  the  given 
equations  and  the  numerator  is  what  the  denominator  becomes 
when  the  right  hand  members  are  put  in  place  of  the  coefficients 
of  the  <7A (?;////!■  -a'/iose  value  i^  soiio/il. 

29.  Another  Method. 
Form  the   determinant 

I  (rt,.i-f- /',.)'+  .  .  +/,'■—'",)    ''',    i\  ■  ■  A 

i  (a.,x-\-h.,y-\-   .    .  +l.,v-7n.,)     A,     C  .    •  L, 


I  (a^x+bj'-\-   .    .      Iv—m^^)     b^^    r^  .    .  /„ 

Each  element  in  the  first  column  is  formed  by  transposing  the 

right-hand  members  of  the   given  equations,  and   hence   each  of 

these  elements  equals  zero,  therefore  the  determinant  itself  equals 

zero.      Now  each  element  in  the  first  column  is    expressed  as   the 


70  ADVANCED     ALGEBRA. 

algebraic  sura  of  ;/-|- I  quantities,  hence    the    deterrainant    can    be 
expressed  as  the  sum  of  «+  i  determinants.     Whence 
I  a^.v  b^  r,  .  .  /,  I         I  b^y  b^  c\  .  .  l^\  I  w,   b^  l\  .  .  /, 

I     ^''2-^"     ^-l     ^l-     ■    ^A     _|_      I     '''•-'J'     ''•'2     "^2  •     •    "^2     I        1  I     '^^2      ^2     ^-l-     •    ^' 

1  ax  b,^  r„ ..".  /,  I         I  b^j  b]  c[  .  .  /„  i  I  m„  'b,^  r,. ."  .'  /„  I 

All  these  determinants,  except  the  first  and  last,  vanish;  for  after 
taking  out  the  common  factor  from  the  first  column  we  have  left 
a  determinant  with  two  identical  columns.  Moreover  after  taking 
out  the  common  factor  x  from  the  first  column  of  the  first  deter- 
minant we  have  left  the  determinant  J  and  the  last  determinant  is 
evidently  what  we  have  called  J,;  hence  J.v — -11=0,  orJr=J,, 
or 


the  same  as  before.     Similarly  the  other  unknown  quantities  may 
be  found. 

30.  If  the  determinant  called  J  should  equal  zero,  we  cannot 
obtain  a  definite,  finite  set  of  values  of  the  unknown  quantities. 
If  all  the  numerators  are  also  zero,  the  unknown  quantities  are 
indeterminate  and  the  equations  given  are  not  independent.  If 
however,  some  of  the  numerators  are  not  zero  then  the  equations 
cannot  be  satisfied  by  any  finite  set  of  values,  or  the  equations  are 
said  to  be  incompatible. 
Thus  the  equations 

2.1- -f  j'4-    55-=  1 9. 

3.r+2_r4-    42'=  19. 

7  r+4j'+ 125=49. 
are  independent  and  compatible  and  therefore  form  a  solvable   set 
of    equations.        In    this    example     -1= — 9      J,  =  — 9,     J,=  — 18, 
J.,= — 27,  whence  .1=  I,  r=2,  z=2)- 
The  equations 

2r+  _)'+    52-=  1 9 

3-V+2i'-f    4z=ig 

7.1- 4- 4_v+ 145=57 


DETERMINANTS. 


are  compatible  but  are  not  independent,  the  third  being  derived 
from  the  other  two  by  adding  the  second  to  twice  the  first. 
In  this  example  J=o,   -l,=o,    -l.,=o,    -';i=o,  so  that  the  values  of 

each  unknown  quantity  assumes  the  indeterminate  form     . 

o 

The  equations 

2.r+    1'+   5.3-=  1 9 

5x-\-2j'-{-   45-=  1 9 

7X+4J/+ 145=50 

are  incompatible  with  one  another.     If  we  add  the  second  to  twice 

the  first  we  get 

7.r+4j'+ 145=57 
but  this  contradicts   the  third  equation.      In  this  example  J=o, 
-I, =42.    J.,=  — 49,  J.^=— 7,  so  that  no  finite  values  of  x,  y,  2  sat- 
isfy the  equations. 

30       IvCt  us  now  take  n  equations  of  the  first  degree  contain- 
ing «  — I  unknown  quantities: 

a.x  +b,v-\-  c,3-\-  ....   4-/,=o 


^n^'+^ny+    ^n^+      ....      +/   =0 

The  absolute  term  is  written  on  the  left-hand  side  of  the  equations 
because  it  is  better  for  the  method  here  pursued.  It  is  to  be  noticed 
that  the  number  of  equations  is  one  more  than  enough  to  enable 
us  to  find  the  values  of  the  unknown  quantities.  Represent  the 
deteirainant  of  the  known  quantities  by  J,  whence 
\a,b,c^  .  .  .  .  /, 
i_     ^..  b.,  c^    .    .    .    .  l, 


I    '^n   ^n  ^n      ■       ■        ■       ■    ^„ 

Now  if  we  add  to  the  last  column  x  times  the  first,  jy  times  the 
second,  z  tiraes  the  third,  and  so  on,  the  determinant  is  not  changed 
in  value;    therefore 

b^  c^  .    .  {a^x-j-b^y-\-c^3-{-    .    .-(-/,) 


b^^  r,  .    .  {a^x+b^^y-\-c^z-\- 


■0 


72  ADVANCED     ALGEBRA. 

But  from  the  given  equations  it  is  evident  that  every  element  in 
the  last  column  equals  zero.      Therefore 

J=o. 

We  have  tacitly  assumed  that  the  given  equations  were  compat- 
ible with  one  another  and  have  shown  that  the  determinant  equals 
zero;  whence  the  following  theroem:  If  n  equations  of  the  first 
degree  containing  n—\  unknown  quantities  are  compatible  with 
one  another  the  determinant  of  the  known  numbers  equals  zero. 
But  this  determinant  may  be  zero  ^^hen  the  equations  are  incom- 
patible,  as  in  the  set: 

2.1-1-   J'-f-     45- 16  =  0 

3,1--!-    J'-f     22 11=0 

8X-I-3V+    8.S-— 38  =  0 
7X-|-3T'-|-  \oz—  40=0 


Here  J= 


Subtracting  twice  the  second  row  from  the  third  we  have  three 
identical  rows,  hence  J=o.  But  if  two  times  the  first  equation  be 
added  to  the  second  we  get  7-i"+3)'-(- io5 — 43^0,  which  contra- 
dicts the  fourth  equation,  whence  the  .system  is  incompatible. 

Let  us  see  if  we  can  tell  when  several  equations  are  compat- 
ible. For  the  sake  of  definiteness  in  language  let  us  take  four 
equations, 

a.^x-\-  b.^y-\-  c,z-\-  d.,=o 
a.p:  -f  k.^  y  -\-  c.z  -{-d.=o 
a  ^x-\-  .\y-\-  r^z+ d  =0 

and  suppo.se  three  of  the.se,  as  for  instance  the  last  three,  are  in- 
dependent and  compatible.  Then  from  these  three  we  find  the 
value  of  .r,  )',  z  to  be 


2    I      4 

—  16 

3    I      2 

—  1 1 

8   3     8 

-38 

7  3   10 

—40 

DETERMINANTS. 


73 


<  K  c. 

\a.,-d,c.,_ 

a,b,-d. 

-d.,  b.  c. 

a,-d,c. 

a,K-d. 

|-<  b^  c, 

l«4-<^. 

«4  b,  -d, 

.     'V  = 

«. 

^ 

^2 

a. 

b., 

^3 

1 

a^ 

b^ 

c^ 

'  a.,  b.^  c^  \  <^-,  b.,  r, 

a^  b^  c^  '  «.   b,  <r. 

Changing  numerators  so  that  the  column  of  a'' s  shall  be  the  last 
column  in  each  case  and  taking  out  the  factor  — i  if  it  occurs 
we  y-et: 


b,  c,  d., 

fl'j  c^  d.. 

a.^  b,  d., 

b,  r,  d., 

«3    ^:i    d.. 

a,  b,  d. 

b,  c.  d, 

a,  c.  d^ 

^4  b,  d. 

^.,  b„  c, 

a,  b.,  c, 

,     ,i 

a,  b^  c. 

a.,  /'.,  r.| 

a,  b..  r. 

a.^  b,  c. 

^4  b^  c^ 

«,   b,  c^ 

a^  b^  c^ 

These  determinants  are  easily  seen  to  be  minors  of 

I  «.  b,  r.  d,  1 

I  *^i  b-t  i\  ^2  I 

I  «:,  b.,  c.^  d.^  I 

I  «4  b,  r,  <  I 

Representing  this  by  J  ,  we  may   write   according   to   a  previous 
notation, 


Now  if  these  values  be  substituted  in  the  first  of  the   given    equa- 
tions we  get 


+ 


<'',J/ 


r,J, 


-J<A 


+    «', 


but  this  is  evidently  J;  therefore  J=o.     Hence  if  the  values  of  x, 
y,  z  that  satisfy  the  last  three  equations  also  satisfy  the  first  then 


74  ADVANCED     ALGEBRA. 

-"=o,  but  if  the  values  of  x,  _r,  z  found  do  not  satisfy  the  first  equa- 
tion then  J  is  zero. 

Our  conclusion  may  now  be  stated    in    general    language   as 
follows:     If  we    have    n    linear    equations   containing    yt — i   un 
known  quantities  and  if  some  n-  i  of  these  equations  are  indepen- 
.  dent  and  compatible  then  the  determinant  of  the  known  numbers 
equated  to  zero  shows  that  the  n  equations  are  compatible. 

30.     If  we  are  given    four    homogeneous^    equations   of  the 
first  degree  containing  four  unknown  quantities,  as 


X      _y       z 
we    may       divide      by       zc    and     then   consider  -,      -,      - 

It.'         7f         w 

the  unknown  quantities  and  we  have  the  case  previously  con- 
sidered. Hence  if  we  have  n  homogentious  equations  of  the  first 
degree  containing  ;«  unknown  quantities  and  some  ;/ — ^i  of  these 
equations  are  independent  and  compatible,  the  determinant  of  the 
coefficients  equated  to  zero  shows  that  the  //  equations  are  com  - 
patible;  or  thus — if  the  determinant  of  the  coefficients  equals  zero 
the  equations  are  compatible  provided  some  n — -i  of  them  are  in- 
dependent and  compatible. 

Homogeneous  equations  may  always  be  satisfied  by  making 
each  of  the  unknown  quantities  zero,  but  this  solution  is  ex- 
cluded and  the  determinants  of  the  coefficients  equal  to  zero  shows 
that  the  equations  are  satisfied  b}-  values  other  than  zero  provided 
some  n — i  of  the  equations  are  independent  and  compatible. 

In  the  apphcations  of  this  theorem  (which  are  numerous)  it  is 
as.suraed  without  statement  that  some  w-i  of  the  equations  are  inde- 
pendent and  compatible  and  the  usual  statement  is  that  if  the  deter- 
minant equals  zero  the  equations  are  compatible. 


*  The  word  "homogeneous"  is  here  used  in  its  strict  sense, meaning  an  equation  whose 
terms  are  of  the  same  degree  and  Zf/VA  «oaA.?o/«/f /crw.  Thus  u  i-+Av--o  is  homogeneous, 
but  ax~by  ^c^=j)  is  uol  homogeneous. 


vA  \ 


ivi30e056 


(M 


THE  UNIVERSITY  OF  CALIFORNIA  LIBRARY 


;-  ;^ 


i^ 


-.  ?"»    -«- . ' 


